Big Ideas Math Geometry Answers Chapter 10 Circles

Big Ideas Math Book Geometry Chapter 10 Circles Answers are provided here. Students who have been looking for the BIM Geometry Chapter 10 Circles Answers can read the following sections. The high school students can find a direct link to download Big Ideas Math Geometry Answers Chapter 10 Circles pdf for free of cost. With the help of this answer key, you can prepare well for the exam.

Big Ideas Math Book Geometry Answer Key Chapter 10 Circles

The different chapters included in Big Ideas Math Geometry Solutions are Lines and Segments That Intersect Circles, Finding Arc Measures, Inscribed Angles and Polygons, Angle Relationships in Circles, Segment Relationships in Circles, Circles in the Coordinate Plane, and Using Chords. Students have to practise all the questions from Big Ideas Math Textbook Geometry Chapter 10 Circles.

This Big Ideas Math Book Geometry Answer Key Chapter 10 Circles helps the students while doing the assignments. Get the solutions for all the questions through the quick links provided in the following sections. Test your skills through performance task, chapter review, and maintaining mathematical proficiency.

Circles Maintaining Mathematical Proficiency

Find the Product.

Question 1.
(x + 7) (x + 4)

Answer:
(x + 7) (x + 4) = x² + 14x + 28

Explanation:
(x + 7) (x + 4) = x(x + 7) + 7(x + 4)
= x² + 7x + 7x + 28
= x² + 14x + 28

Question 2.
(a + 1) (a – 5)

Answer:
(a + 1) (a – 5) = a² – 4a – 5

Explanation:
(a + 1) (a – 5) = a(a – 5) + 1(a – 5)
= a² – 5a + a – 5
= a² – 4a – 5

Question 3.
(q – 9) (3q – 4)

Answer:
(q – 9) (3q – 4) = 3q² – 31q + 32

Explanation:
(q – 9) (3q – 4) = q(3q – 4) – 9(3q – 4)Exercise 10.3 Using Chords

= 3q² – 4q – 27q + 32
= 3q² – 31q + 32

Question 4.
(2v – 7) (5v + 1)

Answer:
(2v – 7) (5v + 1) = 10v² – 33v – 7

Explanation:
(2v – 7) (5v + 1) = 2v(5v + 1)- 7(5v + 1)
= 10v² + 2v – 35v – 7
= 10v² – 33v – 7

Question 5.
(4h + 3) (2 + h)

Answer:
(4h + 3) (2 + h) = 4h² + 11h + 6

Explanation:
(4h + 3) (2 + h) = 4h(2 + h) + 3(2 + h)
= 8h + 4h² + 6 + 3h
= 4h² + 11h + 6

Question 6.
(8 – 6b) (5 – 3b)

Answer:
(8 – 6b) (5 – 3b) = 18b² – 54b + 40

Explanation:
(8 – 6b) (5 – 3b) = 8(5 – 3b) – 6b(5 – 3b)
= 40 – 24b – 30b + 18b²
= 18b² – 54b + 40

Solve the equation by completing the square. Round your answer to the nearest hundredth, if necessary.

Question 7.
x2 – 2x = 5

Answer:
The solutions are x = √6 + 1, x = 1 – √6

Explanation:
x² – 2x = 5
x² – 2x + 1² = 5 + 1²
(x – 1)² = 6
x – 1 = ±√6
x = ±√6 + 1
The solutions are x = √6 + 1, x = -√6 + 1

Question 8.
r2 + 10r = -7

Answer:
The solutions are r = √18 – 5, r = 5 – √18

Explanation:
r2 + 10r = -7
r² + 10r + 5² = -7 + 5²
(r + 5)² = -7 + 25 = 18
r + 5 = ±√18
r = ±√18 – 5
The solutions are r = √18 – 5, r = 5 – √18

Question 9.
w2 – 8w = 9

Answer:
The solutions are w = 9, w = -1

Explanation:
w2 – 8w = 9
w2 – 8w + 4² = 9 + 4²
(w – 4)² = 9 + 16 = 25
w – 4 = ±5
w = 5 + 4, w = -5 + 4
w = 9, w = -1
The solutions are w = 9, w = -1

Question 10.
p2 + 10p – 4 = 0

Answer:
The solutions are p = √29 – 5, p = 5 – √29

Explanation:
p2 + 10p = 4
p² + 10p + 5² = 4 + 5²
(p + 5)² = 4 + 25
(p + 5)² = 29
p + 5 = ±√29
p = ±√29 – 5
The solutions are p = √29 – 5, p = 5 – √29

Question 11.
k2 – 4k – 7 = 0

Answer:
The solutions are k = √11 + 2, k = 2 – √11

Explanation:
k² – 4k= 7
k² – 4k + 2² = 7 + 4
(k – 2)² = 11
k – 2 = ±√11
k = √11 + 2, k = 2 – √11
The solutions are k = √11 + 2, k = 2 – √11

Question 12.
– z2 + 2z = 1

Answer:
The solutions are z = 1

Explanation:
-z² + 2z = 1
z² – 2z = -1
z² – 2z + 1 = -1 + 1
(z – 1)² = 0
z = 1
The solutions are z = 1

Question 13.
ABSTRACT REASONING
write an expression that represents the product of two consecutive positive odd integers. Explain your reasoning.

Answer:
Let us take two consecutive odd integers are x and (x + 2)
The product of two consecutive odd integers is x • (x + 2)
x(x + 2) = x² + 2x

Circles Mathematical Practices

Monitoring progress

Let ⊙A, ⊙B, and ⊙C consist of points that are 3 units from the centers.

Circles Maintaining Mathematical Proficiency Find the Product. Question 1. (x + 7) (x + 4) Answer: Question 2. (a + 1) (a - 5) Answer: Question 3. (q - 9) (3q - 4) Answer: Question 4. (2v - 7) (5v + 1) Answer: Question 5. (4h + 3) (2 + h) Answer: Question 6. (8 - 6b) (5 - 3b) Answer: Solve the equation by completing the square. Round your answer to the nearest hundredth, if necessary. Question 7. x<sup>2</sup> - 2x = 5 Answer: Question 8. r<sup>2</sup> + 10r = -7 Answer: Question 9. w<sup>2</sup> - 8w = 9 Answer: Question 10. p<sup>2</sup> + 10p - 4 = 0 Answer: Question 11. k<sup>2</sup> - 4k - 7 = 0 Answer: Question 12. - z<sup>2</sup> + 2z = 1 Answer: Question 13. ABSTRACT REASONING write an expression that represents the product of two consecutive positive odd integers. Explain your reasoning. Answer: Circles Mathematical Practices Monitoring progress Let ⊙A, ⊙B, and ⊙C consist of points that are 3 units from the centers. im - 1 Question 1. Draw ⊙C so that it passes through points A and B in the figure at the right. Explain your reasoning. Answer: Question 2. Draw ⊙A, ⊙B, and OC so that each is tangent to the other two. Draw a larger circle, ⊙D, that is tangent to each of the other three circles. Is the distance from point D to a point on ⊙D levss than, greater than, or equal to 6? Explain. Answer:

Question 1.
Draw ⊙C so that it passes through points A and B in the figure at the right. Explain your reasoning.

Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 1

Question 2.
Draw ⊙A, ⊙B, and OC so that each is tangent to the other two. Draw a larger circle, ⊙D, that is tangent to each of the other three circles. Is the distance from point D to a point on ⊙D less than, greater than, or equal to 6? Explain.

Answer:

10.1 Lines and Segments that Intersect Circles

Exploration 1

Lines and Line Segments That Intersect Circles

Big Ideas Math Geometry Answers Chapter 10 Circles 2

Work with a partner: The drawing at the right shows five lines or segments that intersect a circle. Use the relationships shown to write a definition for each type of line or segment. Then use the Internet or some other resource to verify your definitions.
Chord: _________________
Secant: _________________
Tangent: _________________
Radius: _________________
Diameter: _________________

Answer:
Chord: A chord of a circle is a straight line segment whose endpoints both lie on a circular arc.
Secant: A straight line that intersects a circle in two points is called a secant line.
Tangent: Tangent line is a line that intersects a curved line at exactly one point.
Radius: It is the distance from the centre of the circle to any point on the circle.
Diameter: It the straight that joins two points on the circle and passes through the centre of the circle.

Exploration 2

Using String to Draw a Circle

Work with a partner: Use two pencils, a piece of string, and a piece of paper.

a. Tie the two ends of the piece of string loosely around the two pencils.
Answer:

b. Anchor one pencil of the paper at the center of the circle. Use the other pencil to draw a circle around the anchor point while using slight pressure to keep the string taut. Do not let the string wind around either pencil.
Big Ideas Math Geometry Answers Chapter 10 Circles 3
Answer:

c. Explain how the distance between the two pencil points as you draw the circle is related to two of the lines or line segments you defined in Exploration 1.
REASONING ABSTRACTLY
To be proficient in math, you need to know and flexibly use different properties of operations and objects.
Answer:

Communicate Your Answer

Question 3.
What are the definitions of the lines and segments that intersect a circle?
Answer:

Question 4.
Of the five types of lines and segments in Exploration 1, which one is a subset of another? Explain.
Answer:

Question 5.
Explain how to draw a circle with a diameter of 8 inches.
Answer:

Lesson 10.1 Lines and Segments that Intersect Circles

Monitoring progress

Question 1.
In Example 1, What word best describes \(\overline{A G}\)? \(\overline{C B}\)?

Answer:
\(\overline{A G}\) is secant because it is a line that intersects the circle at two points.
\(\overline{C B}\) is the radius as it is the distance from the centre to the point of a circle.

Question 2.
In Example 1, name a tangent and a tangent segment.

Answer:
\(\overline{D E}\) is the tangent of the circle
\(\overline{D E}\) is the tangent segment of the circle.

Tell how many common tangents the circles have and draw them. State whether the tangents are external tangents or internal tangents.

Question 3.
Big Ideas Math Geometry Answers Chapter 10 Circles 4

Answer:
4 tangents.
Big Ideas Math Geometry Answers Chapter 10 Circles 2
A tangent is a line segment that intersects the circle at exactly one point. Internal tangents are the lines that intersect the segments joining the centres of two circles. External tangents are the lines that do not cross the segment joining the centres of the circles.
Blue lines represent the external tangents and red lines represent the internal tangents.

Question 4.
Big Ideas Math Geometry Answers Chapter 10 Circles 5

Answer:
One tangent.
Big Ideas Math Geometry Answers Chapter 10 Circles 3
One external tangent.

Question 5.
Big Ideas Math Geometry Answers Chapter 10 Circles 6

Answer:
No tangent.
It is not possible to draw a common tangent for this type of circle.

Question 6.
Is \(\overline{D E}\) tangent to ⊙C?
Big Ideas Math Geometry Answers Chapter 10 Circles 7

Answer:
Use the converse of Pythagorean theorem i.e 2² = 3² + 4²
4 = 9 + 16
By the tangent line to the circle theorem, \(\overline{D E}\) is not a tangent to ⊙C

Question 7.
\(\overline{S T}\) is tangent to ⊙Q.
Find the radius of ⊙Q.
Big Ideas Math Geometry Answers Chapter 10 Circles 8

Answer:
The radius of ⊙Q is 7 units.

Explanation:
By using the Pythagorean theorem
(18 + r)² = r² + 24²
324 + 36r + r² = r² + 576
36r = 576 – 324
36r = 252
r = 7

Question 8.
Points M and N are points of tangency. Find the value(s) of x.
Big Ideas Math Geometry Answers Chapter 10 Circles 9

Answer:
The values of x are 3 or -3.

Explanation:
x² = 9
x = ±3

Exercise 10.1 Lines and Segments that Intersect Circles

Vocabulary and Core Concept Check

Question 1.
WRITING
How are chords and secants alike? How are they different?
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.1 Ans 1

Question 2.
WRITING
Explain how you can determine from the context whether the words radius and diameter are referring to segments or lengths.

Answer:
Radius and diameter are the lengths of the line segments that pass through the centre of a circle. Radius is half of the diameter.

Question 3.
COMPLETE THE SENTENCE
Coplanar circles that have a common center are called ____________ .
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.1 Ans 3

Question 4.
WHICH ONE DOESNT BELONG?
Which segment does not belong with the other three? Explain your reasoning.
chord radius tangent diameter

Answer:
A chord, a radius and a diamter are segments and they intersect a circle in two points. A tangent is a line that intersects a circle at one point.

Monitoring Progress and Modeling with Mathematics

In Exercises 5 – 10, use the diagram.

Big Ideas Math Geometry Answers Chapter 10 Circles 10

Question 5.
Name the circle.
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.1 Ans 5

Question 6.
Name two radii.

Answer:
The name of the two radii is CD and AC.

Question 7.
Name two chords.
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.1 Ans 7

Question 8.
Name a diameter.

Answer:
The name of diameter is AD

Question 9.
Name a secant.
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.1 Ans 9

Question 10.
Name a tangent and a point of tangency

Answer:
GE is the tangent and F is the point of tangency.

In Exercises 11 – 14, copy the diagram. Tell how many common tangents the circles have and draw them.

Question 11.
Big Ideas Math Geometry Answers Chapter 10 Circles 11
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.1 Ans 11

Question 12.
Big Ideas Math Geometry Answers Chapter 10 Circles 12

Answer:
No common tangent because two circles do not intersect at one point.

Question 13.
Big Ideas Math Geometry Answers Chapter 10 Circles 13
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.1 Ans 13

Question 14.
Big Ideas Math Geometry Answers Chapter 10 Circles 14

Answer:
One common tangent.
Big Ideas Math Geometry Answers Chapter 10 Circles 3

In Exercises 15 – 18, tell whether the common tangent is internal or external.

Question 15.
Big Ideas Math Geometry Answers Chapter 10 Circles 15
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.1 Ans 15

Question 16.
Big Ideas Math Geometry Answers Chapter 10 Circles 16

Answer:
The common tangent is the internal tangent because it intersects the segment that joins the centres of two circles.

Question 17.
Big Ideas Math Geometry Answers Chapter 10 Circles 17
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.1 Ans 17

Question 18.
Big Ideas Math Geometry Answers Chapter 10 Circles 18
Answer:
The common tangent is the internal tangent because it intersects the segment that joins the centres of two circles.

In Exercises 19 – 22, tell whether \(\overline{A B}\) is tangent to ⊙C. Explain your reasoning.

Question 19.
Big Ideas Math Geometry Answers Chapter 10 Circles 19
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.1 Ans 19

Question 20.
Big Ideas Math Geometry Answers Chapter 10 Circles 20

Answer:
Use the converse of the Pythagorean theorem
18² _____________ 15² + 9²
324 _____________ 225 + 81
324 ≠ 304
△ ACB is not a right angled triangle.
So, \(\overline{A B}\) is not tangent to ⊙C at B.

Question 21.
Big Ideas Math Geometry Answers Chapter 10 Circles 21
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.1 Ans 21

Question 22.
Big Ideas Math Geometry Answers Chapter 10 Circles 22

Answer:
Use the converse of the Pythagorean theorem
8² _____________ 12² + 16²
64 _____________ 144 + 256
64 ≠ 400
△ ACB is not a right angled triangle.
So, \(\overline{A B}\) is not tangent to ⊙C at B.

In Exercises 23 – 26, point B is a point of tangency. Find the radius r of ⊙C.

Question 23.
Big Ideas Math Geometry Answers Chapter 10 Circles 23
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.1 Ans 23

Question 24.
Big Ideas Math Geometry Answers Chapter 10 Circles 24

Answer:
(r + 6)² = r² + 9²
r² + 12r + 36 = r² + 81
12r = 81 – 36
12r = 45
r = \(\frac { 15 }{ 4 } \)
Therefore, the radius of ⊙C is \(\frac { 15 }{ 4 } \)

Question 25.
Big Ideas Math Geometry Answers Chapter 10 Circles 25
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.1 Ans 25

Question 26.
Big Ideas Math Geometry Answers Chapter 10 Circles 26

Answer:
(r + 18)² = r² + 30²
r² + 36r + 324 = r² + 900
36r = 900 – 324
36r = 576
r = 16
Therefore, the radius of ⊙C is 16

CONSTRUCTION
In Exercises 27 and 28. construct ⊙C with the given radius and point A outside of ⊙C. Then construct a line tangent to ⊙C that passes through A.

Question 27.
r = 2 in.
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.1 Ans 27

Question 28.
r = 4.5 cm

Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 5

In Exercises 29 – 32, points B and D are points of tangency. Find the value(s) of x.

Question 29.
Big Ideas Math Geometry Answers Chapter 10 Circles 27
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.1 Ans 29

Question 30.
Big Ideas Math Geometry Answers Chapter 10 Circles 28

Answer:
3x + 10 = 7x – 6
7x – 3x = 10 + 6
4x = 16
x = 4

Question 31.
Big Ideas Math Geometry Answers Chapter 10 Circles 29
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.1 Ans 31

Question 32.
Big Ideas Math Geometry Answers Chapter 10 Circles 30

Answer:
2x + 5 = 3x² + 2x – 7
3x² = 5 + 7
3x² = 12
x² = 4
x = ±2

Question 33.
ERROR ANALYSIS
Describe and correct the error in determining whether \(\overline{X Y}\) is tangent to ⊙Z.
Big Ideas Math Geometry Answers Chapter 10 Circles 31
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.1 Ans 33

Question 34.
ERROR ANALYSIS
Describe and correct the error in finding the radius of ⊙T.
Big Ideas Math Geometry Answers Chapter 10 Circles 32

Answer:
39² = 36² + 15²
So, 15 is the diameter.
The radius is \(\frac { 15 }{ 2 } \).

Question 35.
ABSTRACT REASONING
For a point outside of a circle, how many lines exist tangent to the circle that pass through the point? How many such lines exist for a point on the circle? inside the circle? Explain your reasoning.
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.1 Ans 35

Question 36.
CRITICAL THINKING
When will two lines tangent to the same circle not intersect? Justify your answer.

Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 6
Using tangent line to circle theorem, it follow that the angle between tangent and radius is a right angle. Let’s draw these tangents at the two ends of the same diameter. We can observe a diameter AD like a transverzal of these tangents.

Question 37.
USING STRUCTURE
Each side of quadrilateral TVWX is tangent to ⊙Y. Find the perimeter of the quadrilateral.
Big Ideas Math Geometry Answers Chapter 10 Circles 33
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.1 Ans 37

Question 38.
LOGIC
In ⊙C, radii \(\overline{C A}\) and \(\overline{C B}\) are perpendicular. Big Ideas Math Geometry Answers Chapter 10 Circles 34 are tangent to ⊙C.

a. Sketch ⊙C, \(\overline{C A}\), \(\overline{C B}\), Big Ideas Math Geometry Answers Chapter 10 Circles 34.
Answer:

b. What type of quadrilateral is CADB? Explain your reasoning.
Answer:

Question 39.
MAKING AN ARGUMENT
Two hike paths are tangent to an approximately circular pond. Your class is building a nature trail that begins at the intersection B of the bike paths and runs between the bike paths and over a bridge through the center P of the pond. Your classmate uses the Converse of the Angle Bisector Theorem (Theorem 6.4) to conclude that the trail must bisect the angle formed by the bike paths. Is your classmate correct? Explain your reasoning.
Big Ideas Math Geometry Answers Chapter 10 Circles 35
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.1 Ans 39

Question 40.
MODELING WITH MATHEMATICS
A bicycle chain is pulled tightly so that \(\overline{M N}\) is a common tangent of the gears. Find the distance between the centers of the gears.
Big Ideas Math Geometry Answers Chapter 10 Circles 36

Answer:
height h = 4.3 – 1.8
h = 2.5
x² = MN² + h²
x² = 17.6² + 2.5²
x² = 316.01
x = 17.8
Therefore, the distance between the centre of the gear is 17.8 in.

Question 41.
WRITING
Explain why the diameter of a circle is the longest chord of the circle.
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.1 Ans 41

Question 42.
HOW DO YOU SEE IT?
In the figure, \(\vec{P}\)A is tangent to the dime. \(\vec{P}\)C is tangent to the quarter, and \(\vec{P}\)B is a common internal tangent. How do you know that \(\overline{P A} \cong \overline{P B} \cong \overline{P C}\)
Big Ideas Math Geometry Answers Chapter 10 Circles 37
Answer:

Question 43.
PROOF
In the diagram, \(\overline{R S}\) is a common internal tangent to ⊙A and ⊙B. Prove that \(\frac{\Lambda C}{B C}=\frac{R C}{S C}\)
Big Ideas Math Geometry Answers Chapter 10 Circles 38
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.1 Ans 43.1
Big Ideas Math Geometry Answers Chapter 10 Circles 10.1 Ans 43.2

Question 44.
THOUGHT PROVOKING
A polygon is circumscribed about a circle when every side of the polygon is tangent to the circle. In the diagram. quadrilateral ABCD is circumscribed about ⊙Q. Is it always true that AB + CD = AD + BC? Justify your answer.
Big Ideas Math Geometry Answers Chapter 10 Circles 39
Answer:

Question 45.
MATHEMATICAL CONNECTIONS
Find the values of x and y. Justify your answer.
Big Ideas Math Geometry Answers Chapter 10 Circles 40
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.1 Ans 45

Question 46.
PROVING A THEOREM
Prove the External Tangent Congruence Theorem (Theorem 10.2).
Big Ideas Math Geometry Answers Chapter 10 Circles 41
Given \(\overline{S R}\) and \(\overline{S T}\) are tangent to ⊙P.
Prove \(\overline{S R} \cong \overline{S T}\)

Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 7
∠PRS and ∠PTS are the right angles. So the legs of circles are congruent.
Therefore, \(\overline{S R} \cong \overline{S T}\)

Question 47.
PROVING A THEOREM
Use the diagram to prove each part of the biconditional in the Tangent Line to Circle Theorem (Theorem 10.1 ).
Big Ideas Math Geometry Answers Chapter 10 Circles 42
a. Prove indirectly that if a line is tangent to a circle, then it is perpendicular to a radius. (Hint: If you assume line m is not perpendicular to \(\overline{Q P}\), then the perpendicular segment from point Q to line m must intersect line m at some other point R.)
Ghen Line m is tangent to ⊙Q at point P.
Prove m ⊥ \(\overline{Q P}\)
b. Prove indirectly that if a line is perpendicular to a radius at its endpoint, then the line is tangent to the circle.
Gien m ⊥ \(\overline{Q P}\)
Prove Line m is tangent to ⊙Q.
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.1 Ans 47.1
Big Ideas Math Geometry Answers Chapter 10 Circles 10.1 Ans 47.2

Question 48.
REASONING
In the diagram, AB = AC = 12, BC = 8, and all three segments are Langent to ⊙P. What is the radius of ⊙P? Justify your answer.
Big Ideas Math Geometry Answers Chapter 10 Circles 43

Answer:

Maintaining Mathematical Proficiency

Find the indicated measure.

Question 49.
m∠JKM
Big Ideas Math Geometry Answers Chapter 10 Circles 44
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.1 Ans 49

Question 50.
AB
Big Ideas Math Geometry Answers Chapter 10 Circles 45

Answer:
AC = AB + BC
10 = AB + 7
AB = 10 – 7
AB = 3

10.2 Finding Arc Measures

Exploration 1

Measuring Circular Arcs

Work with a partner: Use dynamic geometry software to find the measure of \(\widehat{B C}\). Verify your answers using trigonometry.

a.
Big Ideas Math Answers Geometry Chapter 10 Circles 46
Points
A(0, 0)
B(5, 0)
C(4, 3)

Answer:
30 degrees

b.
Big Ideas Math Answers Geometry Chapter 10 Circles 47
Points
A(0, 0)
B(5, 0)
C(3, 4)

Answer:
60 degrees

c.
Big Ideas Math Answers Geometry Chapter 10 Circles 48
Points
A(0, 0)
B(4, 3)
C(3, 4)

Answer:
15 degrees

d.
Big Ideas Math Answers Geometry Chapter 10 Circles 49
Points
A(0, 0)
B(4, 3)
C(- 4, 3)

Answer:
90 degrees

Communicate Your Answer

Question 2.
How are circular arcs measured?
Answer:

Question 3.
Use dynamic geometry software to draw a circular arc with the given measure.
USING TOOLS STRATEGICALLY
To be proficient in math, you need to use technological tools to explore and deepen your understanding of concepts.
a. 30°

Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 8

b. 45°

Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 9

c. 60°
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10

d. 90°

Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 11

Lesson 10.2 Finding Arc Measures

Monitoring Progress

Identify the given arc as a major arc, minor arc, or semicircle. Then find the measure of the arc.

Big Ideas Math Answers Geometry Chapter 10 Circles 50

Question 1.
\(\widehat{T Q}\)

Answer:
\(\widehat{T Q}\) is a minor arc.
\(\widehat{T Q}\) = 120°

Question 2.
\(\widehat{Q R T}\)

Answer:
\(\widehat{Q R T}\)

Question 3.
\(\widehat{T Q R}\) is a major arc.
\(\widehat{Q R T}\) = QR + RS + ST
RS = 360° – (60 + 120 + 80)
= 360 – 260 = 100°
So, \(\widehat{Q R T}\) = 60° + 100° + 80°
\(\widehat{Q R T}\) = 240°

Answer:

Question 4.
\(\widehat{Q S}\)

Answer:
\(\widehat{Q S}\) = QR + RS
= 60 + 100 = 160°
Therefore, \(\widehat{Q S}\) = 160° and it is a minor arc.

Question 5.
\(\widehat{T S}\)

Answer:
\(\widehat{T S}\) = 80° and it is a minor arc.

Question 6.
\(\widehat{R S T}\)

Answer:
\(\widehat{R S T}\) = 100 + 80 = 180
Therefore, \(\widehat{R S T}\) = 180° and it is a minor arc.

Tell whether the red arcs are congruent. Explain why or why not.

Question 7.
Big Ideas Math Answers Geometry Chapter 10 Circles 51

Answer:
\(\widehat{A B}\), \(\widehat{C D}\) are congruent as they measure same radius and same arc length.

Question 8.
Big Ideas Math Answers Geometry Chapter 10 Circles 52

Answer:
\(\widehat{M N}\), \(\widehat{P Q}\) are not congruent as they measure different radius.

Exercise 10.2 Finding Arc Measures

Vocabulary and Core Concept Check

Question 1.
VOCABULARY
Copy and complele:
If ∠ACB and ∠DCE are congruent central angles of ⊙C, then \(\widehat{A B}\) and \(\widehat{D E}\) arc.
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.2 Ans 1

Question 2.
WHICH ONE DOESNT BELONG?
Which circle does not belong with the other three? Explain your reasoning.
Big Ideas Math Answers Geometry Chapter 10 Circles 53

Answer:
We know that 1 ft = 12 in
So, the fourth circle does not belong to the other three as its diameter is different.

Monitoring Progress and Modeling with Mathematics

In Exercises 3 – 6, name the red minor arc and find its measure. Then name the blue major arc and find its measure.

Question 3.
Big Ideas Math Answers Geometry Chapter 10 Circles 54
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.2 Ans 3

Question 4.
Big Ideas Math Answers Geometry Chapter 10 Circles 55

Answer:
The minor arc \(\widehat{E F}\) = 68° and major arc \(\widehat{F G E}\) = 360° – 68° = 292°.

Question 5.
Big Ideas Math Answers Geometry Chapter 10 Circles 56
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.2 Ans 5

Question 6.
Big Ideas Math Answers Geometry Chapter 10 Circles 57

Answer:
The minor arc is \(\widehat{M N}\) = 170°, major arc \(\widehat{N P M}\) = 360° – 170° = 190°.

In Exercises 7 – 14. identify the given arc as a major arc, minor arc, or semicircle. Then find the measure of the arc.

Big Ideas Math Answers Geometry Chapter 10 Circles 58

Question 7.
\(\widehat{B C}\)
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.2 Ans 7

Question 8.
\(\widehat{D C}\)

Answer:
\(\widehat{D C}\) is a minor arc and it has a measure of 65°.

Question 9.
\(\widehat{E D}\)
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.2 Ans 9

Question 10.
\(\widehat{A E}\)

Answer:
\(\widehat{A E}\) is a minor arc and it has a measure of 70°.

Question 11.
\(\widehat{E A B}\)
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.2 Ans 11

Question 12.
\(\widehat{A B C}\)

Answer:
\(\widehat{A B C}\) is a semicircle and it has a measure of 180°.

Question 13.
\(\widehat{B A C}\)
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.2 Ans 13

Question 14.
\(\widehat{E B D}\)

Answer:
\(\widehat{E B D}\) is a major arc and it has a measure of 315°.

In Exercises 15 and 16, find the measure of each arc.

Question 15.
Big Ideas Math Answers Geometry Chapter 10 Circles 59

a. \(\widehat{J L}\)

b. \(\widehat{K M}\)

c. \(\widehat{J L M}\)

d. \(\widehat{J M}\)
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.2 Ans 15

Question 16.
Big Ideas Math Answers Geometry Chapter 10 Circles 60

a. \(\widehat{R S}\)

Answer:
\(\widehat{R S}\) = \(\widehat{Q R S}\) – \(\widehat{Q R}\)
= 180 – 42
= 138°
So, \(\widehat{R S}\) = 138°

b. \(\widehat{Q R S}\)

Answer:
\(\widehat{Q R S}\) = 180°

c. \(\widehat{Q S T}\)

Answer:
\(\widehat{Q S T}\) = \(\widehat{Q R S}\) + \(\widehat{S T}\)
= 180 + 42 = 222
So, \(\widehat{Q S T}\) = 222°

d. \(\widehat{Q T}\)

Answer:
\(\widehat{Q T}\) = 360 – (42 + 138 + 42)
= 360 – (222)
= 138°
\(\widehat{Q T}\) = 138°

Question 17.
MODELING WITH MATHEMATICS
A recent survey asked high school students their favorite type of music. The results are shown in the circle graph. Find each indicated arc measure.
Big Ideas Math Answers Geometry Chapter 10 Circles 61
a. m\(\widehat{A E}\)
b. m\(\widehat{A C E}\)
c. m\(\widehat{G D C}\)
d. m\(\widehat{B H C}\)
e. m\(\widehat{F D}\)
f. m\(\widehat{F B D}\)
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.2 Ans 17

Question 18.
ABSTRACT REASONING
The circle graph shows the percentages of students enrolled in fall Sports at a high school. Is it possible to find the measure of each minor arc? If so, find the measure 0f the arc for each category shown. If not, explain why it is not possible.
Big Ideas Math Answers Geometry Chapter 10 Circles 62

Answer:
Soccer angle = 30% of 360 = 108°
Volleyball angle = 15% of 360 = 54°
Cross-country angle = 20% of 360 = 72°
None angle = 15% of 360 = 54°
Football angle = 20% of 360 = 72°

In Exercises 19 – 22, tell whether the red arcs are congruent. Explain why or why not.

Question 19.
Big Ideas Math Answers Geometry Chapter 10 Circles 63
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.2 Ans 19

Question 20.
Big Ideas Math Answers Geometry Chapter 10 Circles 64

Answer:
\(\widehat{L P}\) and \(\widehat{M N}\) are not congruet because they are not in the same circle.

Question 21.
Big Ideas Math Answers Geometry Chapter 10 Circles 65
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.2 Ans 21

Question 22.
Big Ideas Math Answers Geometry Chapter 10 Circles 66

Answer:
\(\widehat{R S Q}\), \(\widehat{F G H}\) are not congruent because those two circles have different radii.

MATHEMATICAL CONNECTIONS
In Exercises 23 and 24. find the value of x. Then find the measure of the red arc.

Question 23.
Big Ideas Math Answers Geometry Chapter 10 Circles 67
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.2 Ans 23

Question 24.
Big Ideas Math Answers Geometry Chapter 10 Circles 68

Answer:
4x + 6x + 7x + 7x = 360
24x = 360
x = 15
m\(\widehat{R S T}\) = 6(15) + 7(15)
= 90 + 105 = 195°
So, m\(\widehat{R S T}\) = 195°

Question 25.
MAKING AN ARGUMENT
Your friend claims that any two arcs with the same measure are similar. Your cousin claims that an two arcs with the same measure are congruent. Who is correct? Explain.
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.2 Ans 25

Question 26.
MAKING AN ARGUMENT
Your friend claims that there is not enough information given to find the value of x. Is your friend correct? Explain your reasoning.
Big Ideas Math Answers Geometry Chapter 10 Circles 69

Answer:
My friend is wrong.
4x + x + x + 4x = 360
10x = 360
x = 36°

Question 27.
ERROR ANALYSIS
Describe and correct the error in naming the red arc.
Big Ideas Math Answers Geometry Chapter 10 Circles 70
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.2 Ans 27

Question 28.
ERROR ANALYSIS
Describe and correct the error in naming congruent arc.
Big Ideas Math Answers Geometry Chapter 10 Circles 71

Answer:
\(\widehat{J K}\), \(\widehat{N P}\) are not congruent because those two arcs are from different circles.

Question 29.
ATTENDING TO PRECISION
Two diameters of ⊙P are \(\widehat{A B}\) and \(\widehat{C D}\). Find m\(\widehat{A C D}\) and m\(\widehat{A C}\) when m\(\widehat{A D}\) = 20°.
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.2 Ans 29

Question 30.
REASONING
In ⊙R, m\(\widehat{A B}\) = 60°, m\(\widehat{B C}\) = 25°. m\(\widehat{C D}\) = 70°, and m\(\widehat{D E}\) = 20°. Find two possible measures of \(\widehat{A E}\).

Answer:
\(\widehat{A E}\) = 360 – (\(\widehat{A B}\) + \(\widehat{B C}\) + \(\widehat{C D}\) + \(\widehat{D E}\))
= 360 – (60 + 25 + 70 + 20)
= 360 – (175)
= 185
\(\widehat{A E}\) = \(\widehat{A B}\) + \(\widehat{B C}\) + \(\widehat{C D}\) + \(\widehat{D E}\)
= 60 + 25 + 70 + 20 = 175
So, the two possibilities of \(\widehat{A E}\) are 185°, 175°

Question 31.
MODELING WITH MATHEMATICS
On a regulation dartboard, the outermost circle is divided into twenty congruent sections. What is the measure of each arc in this circle?
Big Ideas Math Answers Geometry Chapter 10 Circles 72
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.2 Ans 31

Question 32.
MODELING WITH MATHEMATICS
You can use the time zone wheel to find the time in different locations across the world. For example, to find the time in Tokyo when it is 4 P.M. in San Francisco, rotate the small wheel until 4 P.M. and San Francisco line up, as shown. Then look at Tokyo to see that it is 9 A.M. there.
Big Ideas Math Answers Geometry Chapter 10 Circles 73
a. What is the arc measure between each time zone 0n the wheel?

Answer:
As the circle is divided into 24 sectors, each time zone angle = \(\frac { 360 }{ 24 } \) = 15°

b. What is the measure of the minor arc from the Tokyo zone to the Anchorage zone?

Answer:
The measure of the minor arc from the Tokyo zone to the Anchorage zone = 15 + 15 + 15 + 15 + 15 + 15
= 90°

c. If two locations differ by 180° on the wheel, then it is 3 P.M. at one location when it is _________ at the other location.

Answer:
Kuwaiti city.

Question 33.
PROVING A THEOREM
Write a coordinate proof of the Similar Circles Theorem (Theorem 10.5).
Given ⊙O with center O(0, 0) and radius r.
⊙A with center A(a, 0) and radius s
Prove ⊙O ~ ⊙A
Big Ideas Math Answers Geometry Chapter 10 Circles 74
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.2 Ans 33

Question 34.
ABSTRACT REASONING
Is there enough information to tell whether ⊙C ≅ ⊙D? Explain your reasoning.
Big Ideas Math Answers Geometry Chapter 10 Circles 75

Answer:
Both circles ⊙C and ⊙D have the same radius so those circles are congruent.

Question 35.
PROVING A THEOREM
Use the diagram to prove each part of the biconditional in the Congruent Circles Theorem (Theorem 10.3).
Big Ideas Math Geometry Solutions Chapter 10 Circles 297
a. Given \(\overline{A C} \cong \overline{B D}\)
Prove ⊙A ≅ ⊙B
b. Given ⊙A ≅ ⊙B
prove \(\overline{A C} \cong \overline{B D}\)
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.2 Ans 35

Question 36.
HOW DO YOU SEE IT?
Are the circles on the target similar or congruent? Explain your reasoning.
Big Ideas Math Answers Geometry Chapter 10 Circles 76
Answer:

Question 37.
PROVING A THEOREM
Use the diagram to prove each part of the biconditional in the Congruent Central Angles Theorem (Theorem 10.4).
Big Ideas Math Answers Geometry Chapter 10 Circles 77
a. Given ∠ABC ≅ ∠DAE
Prove \(\widehat{B C}\) ≅ \(\widehat{D E}\)
b. Given \(\widehat{B C}\) ≅ \(\widehat{D E}\)
Prove ∠ABC ≅ ∠DAE
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.2 Ans 37

Question 38.
THOUGHT PROVOKING
Write a formula for the length of a circular arc. Justify your answer.

Answer:
The formula to find the length of a circular arc is radius x angle.

Maintaining Mathematical Proficiency

Find the value of x. Tell whether the side lengths form a Pythagorean triple.

Question 39.
Big Ideas Math Answers Geometry Chapter 10 Circles 78
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.2 Ans 39

Question 40.
Big Ideas Math Answers Geometry Chapter 10 Circles 81

Answer:
x² = 13² + 13²
= 169 + 169
= 338
x = 13√2

Question 41
Big Ideas Math Answers Geometry Chapter 10 Circles 79
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.2 Ans 41

Question 42.
Big Ideas Math Answers Geometry Chapter 10 Circles 80

Answer:
14² = x² + 10²
196 = x² + 100
x² = 196 – 100
x² = 96
x = 4√6

10.3 Using Chords

Exploration 1

Drawing Diameters

Work with a partner: Use dynamic geometry software to construct a circle of radius 5 with center at the origin. Draw a diameter that has the given point as an endpoint. Explain how you know that the chord you drew is a diameter.
a. (4, 3)
b. (0, 5)
c. (-3, 4)
d. (-5, 0)

Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 12

Exploration 2

Writing a Conjecture about Chords

Work with a partner. Use dynamic geometry software to construct a chord \(\overline{B C}\) of a circle A. Construct a chord on the perpendicular bisector of \(\overline{B C}\). What do you notice? Change the original chord and the circle several times. Are your results always the same? Use your results to write a conjecture.
LOOKING FOR STRUCTURE
To be proficient in math, you need to look closely to discern a pattern or structure.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 82
Answer:

Exploration 3

A Chord Perpendicular to a Diameter

Work with a partner. Use dynamic geometry software to construct a diameter \(\overline{B C}\) of a circle A. Then construct a chord \(\overline{D E}\) perpendicular to \(\overline{B C}\) at point F. Find the lengths DF and EF. What do you notice? Change the chord perpendicular to \(\overline{B C}\) and the circle several times. Do you always get the same results? Write a conjecture about a chord that is perpendicular to a diameter of a circle.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 83
Answer:

Communicate Your Answer

Question 4.
What are two ways to determine when a chord is a diameter of a circle?

Answer:
If a chord passes through the centre of the circle, then it is the diameter of a circle.
The longest chord of the circle is the diameter of a circle.

Lesson 10.3 Using Chords

Monitoring Progress

In Exercises 1 and 2, use the diagram of ⊙D.

Big Ideas Math Geometry Answer Key Chapter 10 Circles 84

Question 1.
If m\(\widehat{A B}\) = 110°. find m\(\widehat{B C}\).

Answer:
Because AB and BC are congruent chords in congruent circles, the corresponding minor arcs \(\widehat{A B}\), \(\widehat{B C}\) are congruent by the congruent corresponding chords theorem.
So, \(\widehat{A B}\) = \(\widehat{B C}\)
\(\widehat{B C}\) = 110°

Question 2.
If m\(\widehat{A C}\) = 150° find m\(\widehat{A B}\).

Answer:
\(\widehat{A C}\) = 360 – (\(\widehat{A B}\) + \(\widehat{B C}\))
150 = 360 – 2(\(\widehat{A B}\))
2(\(\widehat{A B}\)) = 360 – 150 = 210
\(\widehat{A B}\) = 105°

In Exercises 3 and 4. find the indicated length or arc measure.

Big Ideas Math Geometry Answer Key Chapter 10 Circles 85

Question 3.
CE

Answer:
CE = 5 + 5
= 10 units

Question 4.
m\(\widehat{C E}\)

Answer:
m\(\widehat{C E}\) = 9x + 180 – x = 180 – 8x
m\(\widehat{C E}\) = 180 – 8x

Question 5.
In the diagram, JK = LM = 24, NP = 3x, and NQ = 7x – 12. Find the radius of ⊙N
Big Ideas Math Geometry Answer Key Chapter 10 Circles 86

Answer:

Exercise 10.3 Using Chords

Vocabulary and Core Concept Check

Question 1.
WRITING
Describe what it means to bisect a chord.
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.3 Ans 1

Question 2.
WRITING
Two chords of a circle are perpendicular and congruent. Does one of them have to be a diameter? Explain your reasoning.

Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 14
Imagine a line segment of length 3 units, AB.
Big Ideas Math Geometry Answers Chapter 10 Circles 15
A second congruent segment of length 3 that is perpendicular to AB called CD.
Big Ideas Math Geometry Answers Chapter 10 Circles 16
Circumscribe both these line segments and note that AB and CD are now chords.
While both chords are perpendicular and congruent, neither chord is a diameter. Thus, it is possible to have two chords of this type with neither one diameter of the circle.

Monitoring Progress and Modeling with Mathematics

In Exercises 3 – 6, find the measure of the red arc or chord in ⊙C.

Question 3.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 87
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.3 Ans 3

Question 4.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 88

Answer:
Arc length = radius x angle
= 5 x 34 = 170

Question 5.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 89
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.3 Ans 5

Question 6.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 90

Answer:
As the two circles radius is the same and the angle is the same so the chord length is 11 units.

In Exercise 7-10, find the value of x.

Question 7.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 91
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.3 Ans 7

Question 8.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 92

Answer:
By the perpendicular bisector theorem RS = ST
x = 40°

Question 9.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 93
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.3 Ans 9

Question 10.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 94

Answer:
5x + 2 = 7x – 12
7x – 5x = 2 + 12
2x = 14
x = 7

Question 11.
ERROR ANALYSIS
Describe and correct the error in reasoning.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 95
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.3 Ans 11

Question 12.
PROBLEM SOLVING
In the cross section of the submarine shown, the control panels are parallel and the same length. Describe a method you can use to find the center of the cross section. Justify your method.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 96

Answer:

In Exercises 13 and 14, determine whether \(\overline{A B}\) is a diameter of the circle. Explain your reasoning.

Question 13.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 97
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.3 Ans 13

Question 14.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 98

Answer:
5² = 3² + x²
25 = 9 + x²
x² = 25 – 9
x = 4
So, AB is not diameter of the circle.

In Exercises 15 and 16, find the radius of ⊙Q.

Question 15.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 99
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.3 Ans 15

Question 16.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 100

Answer:
AD = BC
4x + 4 = 6x – 6
6x – 4x = 4 + 6
2x = 10
x = 5
BC = 6(5) – 6 = 30 – 6 = 24
QC² = 5² + 12²
= 25 + 144 = 169
QC = 13
Therefore, the radius is 13.

Question 17.
PROBLEM SOLVING
An archaeologist finds part of a circular plate. What was the diameter of the plate to the nearest tenth of an inch? Justify your answer.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 101
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.3 Ans 17

Question 18.
HOW DO YOU SEE IT?
What can you conclude from each diagram? Name a theorem that justifies your answer.
a.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 102

Answer:
Perpendicular chord bisector converse theorem.

b.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 103

Answer:
Congruent Corresponding Chords theorem

c.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 104

Answer:
Perpendicular chord bisector theorem

d.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 105

Answer:
Equidistant chords theorem

Question 19.
PROVING A THEOREM
Use the diagram to prove each part of the biconditional in the Congruent Corresponding Chords Theorem (Theorem 10.6).
Big Ideas Math Geometry Answer Key Chapter 10 Circles 106
a. Given \(\overline{A B}\) and \(\overline{C D}\) are congruent chords.
Prove \(\widehat{A B} \cong \widehat{C D}\)
b. Given \(\widehat{A B} \cong \widehat{C D}\)
Prove \(\overline{A B}\) ≅ \(\overline{C D}\)
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.3 Ans 19

Question 20.
MATHEMATICAL CONNECTIONS
In ⊙P, all the arcs shown have integer measures. Show that x must be even.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 109

Answer:

Question 21.
REASONING
In ⊙P. the lengths of the parallel chords are 20, 16, and 12. Find m\(\widehat{A B}\). Explain your reasoning.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 109
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.3 Ans 21

Question 22.
PROVING A THEOREM
Use congruent triangles to prove the Perpendicular Chord Bisector Theorem (Theorem 10.7).
Big Ideas Math Geometry Answer Key Chapter 10 Circles 109
Given \(\overline{E G}\) is a diameter of ⊙L.
\(\overline{E G}\) ⊥ \(\overline{D F}\)
Prove \(\overline{D C}\) ≅ \(\overline{F C}\), \(\widehat{D G} \cong \widehat{F G}\)

Answer:
Let L be the centre of the circle
draw any chord DF on the circle
As DF passes through LG.
The length of DC is the same as FC.

Question 23.
PROVING A THEOREM
Write a proof of the Perpendicular Chord Bisector Converse (Theorem 10.8).
Big Ideas Math Geometry Answer Key Chapter 10 Circles 110
Given \(\overline{Q S}\) is a perpendicular bisector of \(\overline{R T}\).
Prove \(\overline{Q S}\) is a diameter of the circle L.
(Hint: Plot the center L and draw △LPT and △LPR.)
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.3 Ans 23

Question 24.
THOUGHT PROVOKING
Consider two chords that intersect at point P. Do you think that \(\frac{A P}{B P}=\frac{C P}{D P}\)? Justify your answer.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 111

Answer:

Question 25.
PROVING A THEOREM
Use the diagram with the Equidistant Chords Theorem (Theorem 10.9) to prove both parts of the biconditional of this theorem.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 112
\(\overline{A B}\) ≅ \(\overline{C D}\) if and only if EF = EG
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.3 Ans 25

Question 26.
MAKING AN ARGUMENT
A car is designed so that the rear wheel is only partially visible below the body of the car. The bottom edge of the panel is parallel [o the ground. Your friend claims that the point where the tire touches the ground bisects \(\widehat{A B}\). Is your friend correct? Explain your reasoning.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 113

Answer:

Maintaining Mathematical Proficiency

Find the missing interior angle measure.

Question 27.
Quadrilateral JKLW has angle measures m∠J = 32°, m∠K = 25°, and m∠L = 44°. Find m∠M.
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.3 Ans 27

Question 28.
Pentagon PQRST has angle measures m∠P = 85°, m∠Q = 134°, m∠R = 97°, and m∠S =102°.
Find m∠T.

Answer:
The sum of interior angles of a pentagon = 540°
m∠T = 540 – (85 + 134 + 97 + 102)
= 540 – 418 = 122
m∠T = 122°.

10.1 – 10.3 Quiz

In Exercises 1 – 6, use the diagram. (Section 10.1)

Big Ideas Math Geometry Answer Key Chapter 10 Circles 114

Question 1.
Name the circle.

Answer:
The circle has a chord, diameter and tangent.

Question 2.
Name a radius.

Answer:
NP is the radius of the circle.

Question 3.
Name a diameter.

Answer:
KN is the diameter of the circle.

Question 4.
Name a chord.

Answer:
JL is the chord

Question 5.
Name a secant.

Answer:
SN is the secant

Question 6.
Name a tangent.

Answer:
QR is the tangent.

Find the value of x.

Question 7.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 115

Answer:
(9 + x)² = x² + 15²
81 + 18x + x² = x² + 225
18x = 225 – 81
18x = 144
x = 8

Question 8.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 116
Answer:
6x – 3 = 3x + 18
6x – 3x = 18 + 3
3x = 21
x = 7

Identify the given arc as a major arc, minor arc, or semicircle. Then find the measure of the arc.

Big Ideas Math Geometry Answer Key Chapter 10 Circles 117

Question 9.
\(\widehat{A E}\)

Answer:
\(\widehat{A E}\) = 180 – 36
= 144
So, \(\widehat{A E}\) = 144°

Question 10.
\(\widehat{B C}\)

Answer:
\(\widehat{B C}\) = 180 – (67 + 70)
= 180 – 137 = 43
So, \(\widehat{B C}\) = 43°

Question 11.
\(\widehat{A C}\)

Answer:
\(\widehat{A C}\) = 43 + 67 = 110°

Question 12.
\(\widehat{A C D}\)

Answer:
\(\widehat{A C D}\) = 180°

Question 13.
\(\widehat{A C E}\)

Answer:
\(\widehat{A C E}\) = 180 + 36 = 216°

Question 14.
\(\widehat{B E C}\)

Answer:
\(\widehat{B E C}\) = 70 + 36 + 43 = 149°

Tell whether the red arcs are congruent. Explain why or why not.

Question 15.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 118

Answer:
As two chords pass through the centre of the circle. Those two red arcs are congruent.

Question 16.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 119
Answer:
Red arcs are not congruent because the radius of the two circles is different.

Question 17.
Find the measure of the red arc in ⊙Q.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 120

Answer:

Question 18.
In the diagram. AC = FD = 30, PG = x + 5, and PJ = 3x – 1. Find the radius of ⊙P.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 121

Answer:

Question 19.
A circular clock can be divided into 12 congruent sections.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 122

a. Find the measure of each arc in this circle.
Answer:
The measure of each arc = \(\frac { 360 }{ 12 } \) = 30°

b. Find the measure of the minor arc formed by the hour and minute hands when the times is 7:00.
Answer:
When the time is 7:00 the minute hand is at 12 and hour hand is at 7 and so the minor arc is subtended by 12 – 7 = 5 of these sections and so the angle subtended is 30 x 5 = 150°

c. Find a time at which the hour and minute hands form an arc that is congruent to the arc in part (b).
Answer:
A minor arc is equal to 150° can be formed by multiple placements of the hour and the minute hand. One of them can be the time 5:00 when the minute hand is at 12 and the hour hand is at 5.

10.4 Inscribed Angles and Polygons

Exploration 1

Inscribed Angles and Central Angles

work with a partner: Use dynamic geometry software.

Sample
Big Ideas Math Geometry Solutions Chapter 10 Circles 123

a. Construct an inscribed angle in a circle. Then construct the corresponding central angle.
Answer:

b. Measure both angles. How is the inscribed angle related to its intercepted arc?
Answer:

c. Repeat parts (a) and (b) several times. Record your results in a table. Write a conjecture about how an inscribed angle is related to its intercepted arc.
ATTENDING TO PRECISION
To be proficient in math, you need to communicate precisely with others.
Answer:

Exploration 2

A Quadrilateral with Inscribed Angles

work with a partner: Use dynamic geometry software.

Sample
Big Ideas Math Geometry Solutions Chapter 10 Circles 124

a. Construct a quadrilateral with each vertex on a circle.
Answer:

b. Measure all four angles. What relationships do you notice?
Answer:

c. Repeat parts (a) and (b) several times. Record your results in a table. Then write a conjecture that summarizes the data.
Answer:

Communicate Your Answer

Question 3.
How are inscribed angles related to their intercepted arcs? How are the angles of an inscribed quadrilateral related to each other?
Answer:

Question 4.
Quadrilateral EFGH is inscribed in ⊙C. and m ∠ E = 80°. What is m ∠ G? Explain.

Answer:
m ∠ E + m ∠ H = 80 + 80 = 160°
m ∠ E + m ∠ H + m ∠ G + m ∠ F = 360
160° + m ∠ G + m ∠ F = 360
m ∠ G + m ∠ F = 360 – 160 = 200
m ∠ G = 100°

Lesson 10.4 Inscribed Angles and Polygons

Monitoring Progress

Find the measure of the red arc or angle.

Question 1.
Big Ideas Math Geometry Solutions Chapter 10 Circles 125

Answer:
m∠G = \(\frac { 90 }{ 2 } \) = 45°

Question 2.
Big Ideas Math Geometry Solutions Chapter 10 Circles 126

Answer:
\(\widehat{T V}\) = 2 • 38 = 76°

Question 3.
Big Ideas Math Geometry Solutions Chapter 10 Circles 127

Answer:
m∠W = 72°

Find the value of each variable.

Question 4.
Big Ideas Math Geometry Solutions Chapter 10 Circles 128

Answer:
x° = 90°
y° = 180 – (40 + 90) = 180 – 130
y° = 50°

Question 5.
Big Ideas Math Geometry Solutions Chapter 10 Circles 129

Answer:
∠B + ∠D = 180
∠B + 82 = 180
x° = 98°
∠C + ∠A = 180
68 + y° = 180
y° = 112°

Question 6.
Big Ideas Math Geometry Solutions Chapter 10 Circles 130

Answer:
∠S + ∠U = 180°
c + 2c – 6 = 180
3c = 186
c = 62°
∠T + ∠V = 180°
10x + 8x = 180
18x = 180
x = 10°

Question 7.
In Example 5, explain how to find locations where the left side of the statue is all that appears in your camera’s field of vision.
Answer:

Exercise 10.4 Inscribed Angles and Polygons

Vocabulary and Core Concept Check

Question 1.
VOCABULARY
If a circle is circumscribed about a polygon, then the polygon is an ___________ .
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.4 Ans 1

Question 2.
DIFFERENT WORDS, SAME QUESTION
Which is different?
Find “both” answers.
Big Ideas Math Geometry Solutions Chapter 10 Circles 131

Find m∠ABC.
Answer:
m∠ABC = 60°

Find m∠AGC.
Answer:
m∠AGC = 180 – (25 + 25)
= 180 – 50 = 130°

Find m∠AEC.
Answer:
m∠AEC = 180 – (50 + 50)
= 180 – 100 = 80°

Find m∠ADC.
Answer:
m∠ADC = 180 – (25 + 50)
= 180 – 75 = 105°

Monitoring Progress and Modeling with Mathematics

In Exercises 3 – 8, find the indicated measure.

Question 3.
m∠A
Big Ideas Math Geometry Solutions Chapter 10 Circles 132
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.4 Ans 3

Question 4.
m∠G
Big Ideas Math Geometry Solutions Chapter 10 Circles 133
Answer:
m∠G = 360 – (70 + 120)
= 360 – 190 = 170°

Question 5.
m ∠ N
Big Ideas Math Geometry Solutions Chapter 10 Circles 134
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.4 Ans 5

Question 6.
m\(\widehat{R S}\)
Big Ideas Math Geometry Solutions Chapter 10 Circles 135
Answer:
m\(\widehat{R S}\) = 2 • 67 = 134°

Question 7.
m\(\widehat{V U}\)
Big Ideas Math Geometry Solutions Chapter 10 Circles 136
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.4 Ans 7

Question 8.
m\(\widehat{W X}\)
Big Ideas Math Geometry Solutions Chapter 10 Circles 137
Answer:
m\(\widehat{W X}\) = \(\frac { 75 }{ 2 } \) = 37.5

In Exercises 9 and 10, name two pairs of congruent angles.

Question 9.
Big Ideas Math Geometry Solutions Chapter 10 Circles 138
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.4 Ans 9

Question 10.
Big Ideas Math Geometry Solutions Chapter 10 Circles 139
Answer:
m∠W = m∠Z, m∠X = m∠Y

In Exercises 11 and 12, find the measure of the red arc or angle.

Question 11.
Big Ideas Math Geometry Solutions Chapter 10 Circles 140
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.4 Ans 11

Question 12.
Big Ideas Math Geometry Solutions Chapter 10 Circles 141
Answer:
\(\widehat{P S}\) = 2 • 40 = 80

In Exercises 13 – 16, find the value of each variable.

Question 13.
Big Ideas Math Geometry Solutions Chapter 10 Circles 142
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.4 Ans 13

Question 14.
Big Ideas Math Geometry Solutions Chapter 10 Circles 143
Answer:
m∠E + m∠G = 180
m + 60 = 180
m = 120°
m∠D + m∠F = 180
60 + 2k = 180
k = 60°

Question 15.
Big Ideas Math Geometry Solutions Chapter 10 Circles 144
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.4 Ans 15

Question 16.
Big Ideas Math Geometry Solutions Chapter 10 Circles 145
Answer:
3x° = 90°
x° = 30°
2y° + 90° + 34° = 180°
2y° + 124° = 180°
2y° = 56°
y° = 28°

Question 17.
ERROR ANALYSIS
Describe and correct the error in finding m\(\widehat{B C}\).
Big Ideas Math Geometry Solutions Chapter 10 Circles 146
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.4 Ans 17

Question 18.
MODELING WITH MATHEMATICS
A carpenter’s square is an L-shaped tool used to draw right angles. You need to cut a circular piece of wood into two semicircles. How can you use the carpenter’s square to draw a diameter on the circular piece of wood?
Big Ideas Math Geometry Solutions Chapter 10 Circles 147
Answer:
Recall that when a right triangle is inscribed in a circle, then the hypotenuse is the diameter of the circle. Simply use the carpenter’s square to inscribe it into the circle. The hypotenuse formed by both legs of the square should provide a diameter for the circle.

MATHEMATICAL CONNECTIONS
In Exercises 19 – 21, find the values of x and y. Then find the measures of the interior angles of the polygon.

Question 19.
Big Ideas Math Geometry Solutions Chapter 10 Circles 148
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.4 Ans 19

Question 20.
Big Ideas Math Geometry Solutions Chapter 10 Circles 149
Answer:
∠B + ∠C = 180
14x + 4x = 180
18x = 180°
x = 10°
∠A + ∠D = 180
9y + 24y = 180
33y = 180°
y = 5.45°
∠A = 130.9°, ∠B = 40°, ∠C = 140°, ∠D = 49°

Question 21.
Big Ideas Math Geometry Solutions Chapter 10 Circles 150
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.4 Ans 21

Question 22.
MAKING AN ARGUMENT
Your friend claims that ∠PTQ ≅ ∠PSQ ≅ ∠PRQ. Is our friend correct? Explain your reasoning.
Big Ideas Math Geometry Solutions Chapter 10 Circles 151
Answer:
Yes, my friend is correct.
∠PTQ ≅ ∠PSQ ≅ ∠PRQ is correct according to the inscribed angles of a circle theorem.

Question 23.
CONSTRUCTION
Construct an equilateral triangle inscribed in a circle.
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.4 Ans 23

Question 24.
CONSTRUTION
The side length of an inscribed regular hexagon is equal to the radius of the circumscribed circle. Use this fact to construct a regular hexagon inscribed in a circle.

Answer:
As the side length is equal to the radius. Draw a line representing the radius and draw a chord different chords in the form of hexagons of the radius of the circle.

REASONING
In Exercises 25 – 30, determine whether a quadrilateral of the given type can always be inscribed inside a circle. Explain your reasoning.

Question 25.
Square
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.4 Ans 25

Question 26.
rectangle
Answer:
yes, angles are right angles.

Question 27.
parallelogram
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.4 Ans 27

Question 28.
kite
Answer:
No.

Question 29.
rhombus
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.4 Ans 29

Question 30.
isosceles trapezoid
Answer:
Yes, the opposite angles are always supplementary.

Question 31.
MODELING WITH MATHEMATICS
Three moons, A, B, and C, are in the same circular orbit 1,00,000 kilometers above the surface of a planet. The planet is 20,000 kilometers in diameter and m∠ABC = 90°. Draw a diagram of the situation. How far is moon A from moon C?
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.4 Ans 31

Question 32.
MODELING WITH MATHEMATICS
At the movie theater. you want to choose a seat that has the best viewing angle, so that you can be close to the screen and still see the whole screen without moving your eyes. You previously decided that seat F7 has the best viewing angle, but this time someone else is already sitting there. Where else can you sit so that your seat has the same viewing angle as seat F7? Explain.
Big Ideas Math Geometry Solutions Chapter 10 Circles 152
Answer:

Question 33.
WRITING
A right triangle is inscribed in a circle, and the radius of the circle is given. Explain how to find the length of the hypotenuse.
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.4 Ans 33

Question 34.
HOW DO YOU SEE IT?
Let point Y represent your location on the soccer field below. What type of angle is ∠AYB if you stand anywhere on the circle except at point A or point B?
Big Ideas Math Geometry Solutions Chapter 10 Circles 153
Answer:

Question 35.
WRITING
Explain why the diagonals of a rectangle inscribed in a circle are diameters of the circle.
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.4 Ans 35

Question 36.
THOUGHT PROVOKING
The figure shows a circle that is circumscribed about ∆ABC. Is it possible to circumscribe a circle about any triangle? Justify your answer.
Big Ideas Math Geometry Solutions Chapter 10 Circles 154
Answer:
Yes.

Question 37.
PROVING A THEOREM
If an angle is inscribed in ⊙Q. the center Q can be on a side of the inscribed angle, inside the inscribed angle, or outside the inscribed angle. Prove each case of the Measure of an Inscribed Angle Theorem (Theorem 10. 10).

a. Case 1
Given ∠ABC is inscribed in ⊙Q
Let m∠B = x°
Center Q lies on \(\overline{B C}\).
Prove m∠ABC = \(\frac{1}{2}\)m\(\widehat{A C}\)
(Hint: Show that ∆AQB is isosceles. Then write m\(\widehat{A C}\) in terms of x.)
Big Ideas Math Geometry Solutions Chapter 10 Circles 155
b. Case 2
Use the diagram and auxiliary line to write Given and Prove statements for Case 2. Then write a proof
Big Ideas Math Geometry Solutions Chapter 10 Circles 156
c. Case 3
Use the diagram and auxiliary line to write Given and Prove statements for Case 3. Then write a proof.
Big Ideas Math Geometry Solutions Chapter 10 Circles 157
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.4 Ans 37.1
Big Ideas Math Geometry Answers Chapter 10 Circles 10.4 Ans 37.2
Big Ideas Math Geometry Answers Chapter 10 Circles 10.4 Ans 37.3

Question 38.
PROVING A THEOREM
Write a paragraph proof of the Inscribed Angles of a Circle Theorem (Theorem 10.11). First, draw a diagram and write Given and Prove statements.
Answer:
If two inscribed angles of a circle intercept the same arc, then the angles are congruent.

Question 39.
PROVING A THEOREM
The Inscribed Right Triangle Theorem (Theorem 10.12) is written as a conditional statement and its converse. Write a plan for proof for each statement.
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.4 Ans 39

Question 40.
PROVING A THEOREM
Copy and complete the paragraph proof for one part of the Inscribed Quadrilateral Theorem (Theorem 10. 13).
Given ⊙C with inscribed quadrilateral DEFG
Prove m ∠ D + m ∠ F = 180°,
m ∠ E + m ∠ G = 180°
Big Ideas Math Geometry Solutions Chapter 10 Circles 158
By the Arc Addition Postulate (Postulate 10. 1),
m\(\widehat{E F G}\) + ________ = 360° and m\(\widehat{F G D}\) + m\(\widehat{D E F}\) = 360°.
Using the ___________ Theorem. m\(\widehat{E D G}\) = 2m ∠ F, m\(\widehat{E F G}\) = 2m∠D, m\(\widehat{D E F}\) = 2m∠G, and m\(\widehat{F G D}\) = 2m ∠ E. By the Substitution Property of Equality, 2m∠D + ________ = 360°, So _________ . Similarly, __________ .
Answer:
m\(\widehat{E F G}\) + m\(\widehat{E D F}\) = 360° and m\(\widehat{F G D}\) + m\(\widehat{D E F}\) = 360°.
Using the the measure of an inscribed angle Theorem. m\(\widehat{E D G}\) = 2m ∠ F, m\(\widehat{E F G}\) = 2m∠D, m\(\widehat{D E F}\) = 2m∠G, and m\(\widehat{F G D}\) = 2m ∠ E. By the Substitution Property of Equality, 2m∠D + 2m∠G = 360°.

Question 41.
CRITICAL THINKING
In the diagram, ∠C is a right angle. If you draw the smallest possible circle through C tangent to \(\overline{A B}\), the circle will intersect \(\overline{A C}\) at J and \(\overline{B C}\) at K. Find the exact length of \(\overline{J K}\).
Big Ideas Math Geometry Solutions Chapter 10 Circles 159
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.4 Ans 41

Question 42.
CRITICAL THINKING
You are making a circular cutting board. To begin, you glue eight 1-inch boards together, as shown. Then you draw and cut a circle with an 8-inch diameter from the boards.
Big Ideas Math Geometry Solutions Chapter 10 Circles 160
a. \(\overline{F H}\) is a diameter of the circular cutting board. Write a proportion relating GJ and JH. State a theorem in to justify your answer.
Answer:
Each board is 1 inch and FJ spans 6 boards.
\(\overline{F H}\) = 6 inches

b. Find FJ, JH, and GJ. What is the length of the cutting board seam labeled \(\overline{G K}\)?
Answer:
Each board is 1 inch and JH spans 2 boards.
JH = 2 inches
Equation is \(\frac { JH }{ GJ } \) = \(\frac { GJ }{ FJ } \)
\(\frac { 2 }{ GJ } \) = \(\frac { GJ }{ 6 } \)
12 = GJ²
GJ = 2√3
GK = 2(GJ)
GK = 4√3
So, FJ = 6, JH = 2, JG = 2√3, GK = 4√3

Maintaining Mathematical Proficiency

Solve the equation. Check your solution.

Question 43.
3x = 145
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.4 Ans 43

Question 44.
\(\frac{1}{2}\)x = 63
Answer:
x = 63 • 2
x = 126

Question 45.
240 = 2x
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.4 Ans 45

Question 46.
75 = \(\frac{1}{2}\)(x – 30)
Answer:
75 • 2 = x – 30
150 + 30 = x
x = 180

10.5 Angle Relationships in Circles

Exploration 1

Angles Formed by a Chord and Tangent Line

Work with a partner: Use dynamic geometry software.

Sample
Big Ideas Math Answer Key Geometry Chapter 10 Circles 161

a. Construct a chord in a circle. At one of the endpoints of the chord. construct a tangent line to the circle.
Answer:

b. Find the measures of the two angles formed by the chord and the tangent line.
Answer:

c. Find the measures of the two circular arcs determined by the chord.
Answer:

d. Repeat parts (a) – (c) several times. Record your results in a table. Then write a conjecture that summarizes the data.
Answer:

Exploration 2

Angles Formed by Intersecting Chords

Work with a partner: Use dynamic geometry software.

sample
Big Ideas Math Answer Key Geometry Chapter 10 Circles 162

a. Construct two chords that intersect inside a circle.
Answer:

b. Find the measure of one of the angles formed by the intersecting chords.
Answer:

c. Find the measures of the arcs intercepted h the angle in part (b) and its vertical angle. What do you observe?
Answer:

d. Repeat parts (a) – (c) several times. Record your results in a table. Then write a conjecture that summarizes the data.
CONSTRUCTING VIABLE ARGUMENTS
To be proficient in math, you need to understand and use stated assumptions, definitions, and previously established results.
Answer:

Communicate Your Answer

Question 3.
When a chord intersects a tangent line or another chord, what relationships exist among the angles and arcs formed?
Answer:

Question 4.
Line m is tangent to the circle in the figure at the left. Find the measure of ∠1.
Big Ideas Math Answer Key Geometry Chapter 10 Circles 163
Answer:
m∠1 = \(\frac { 1 }{ 2 } \) • 148
m∠1 = 74°

Question 5.
Two chords intersect inside a circle to form a pair of vertical angles with measures of 55°. Find the sum of the measures of the arcs intercepted by the two angles.
Answer:
The sum of the measures of the arcs intercepted by the two angles = \(\frac { 1 }{ 2 } \) • 55
= 27.5

Lesson 10.5 Angle Relationships in Circles

Monitoring Progress

Line m is tangent to the circle. Find the indicated measure.

Question 1.
m ∠ 1
Big Ideas Math Answer Key Geometry Chapter 10 Circles 164
Answer:
m ∠ 1 = \(\frac { 1 }{ 2 } \) • 210
m ∠ 1 = 105°

Question 2.
m\(\widehat{R S T}\)
Big Ideas Math Answer Key Geometry Chapter 10 Circles 165
Answer:
m\(\widehat{R S T}\) = 2 • 98 = 196°
m\(\widehat{R S T}\) = 196°

Question 3.
m\(\widehat{X Y}\)
Big Ideas Math Answer Key Geometry Chapter 10 Circles 166
Answer:
m\(\widehat{X Y}\) = \(\frac { 1 }{ 2 } \) • 80
m\(\widehat{X Y}\) = 40°

Find the value of the variable.

Question 4.
Big Ideas Math Answer Key Geometry Chapter 10 Circles 167
Answer:
y° = \(\frac { 1 }{ 2 } \) • (102 + 95)
= 98.5°

Question 5.
Big Ideas Math Answer Key Geometry Chapter 10 Circles 168
Answer:
a° = 2 • 30° + 44°
= 60° + 44° = 104°
So, a° = 104°.

Find the value of x.

Question 6.
Big Ideas Math Answer Key Geometry Chapter 10 Circles 169
Answer:
x° = 180° – 120°
x° = 60°

Question 7.
Big Ideas Math Answer Key Geometry Chapter 10 Circles 170
Answer:
50° = 180° – x°
x° = 180° – 50°
x° = 130°

Question 8.
You are on top of Mount Rainier on a clear day. You are about 2.73 miles above sea level at point B. Find m\(\widehat{C D}\), which represents the part of Earth that you can see.
Big Ideas Math Answer Key Geometry Chapter 10 Circles 171
Answer:
CB and BD are tangents, CB is perpendicular to AB and CD is perpendicular to AD by the tangent line to circle theorem.
△ABC is similar to △ABD by the hypotenuese leg congruence theorem.
∠CBA is similar to ∠ABD. So, m∠CBA = 74.5°, m∠CBD = 2 • 74.5° = 149°
m∠CBD = 180° – m∠CAD
m∠CBD = 180° – CD
149° = 180° – CD
CD = 31°
The part of earth you can see

Exercise 10.5 Angle Relationships in Circles

Vocabulary and Core Concept Check

Question 1.
COMPLETE THE SENTENCE
Points A, B, C, and D are on a circle, and Big Ideas Math Answer Key Geometry Chapter 10 Circles 172 intersects Big Ideas Math Answer Key Geometry Chapter 10 Circles 173 at point P.
If m∠APC = \(\frac{1}{2}\)(m\(\widehat{B D}\) – m\(\widehat{A C}\)). then point P is _________ the circle.
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.5 Ans 1

Question 2.
WRITING
Explain how to find the measure of a circumscribed angle.
Answer:
A circumscribed angle is the angle made by two intersecting tangent lines to a circle. Draw lines from the circle centre to the point of tangency. The angle between the radius and tangent line is 90°. The sum of angles of a quadrilateral is 360°. Angles between radii and tangent lines is 180°. The angle at two tangent lines meet is circumscribed angle.

Monitoring Progress and Modeling with Mathematics

In Exercises 3 – 6, line t is tangent to the circle. Find the indicated measure.

Question 3.
m\(\widehat{A B}\)
Big Ideas Math Answer Key Geometry Chapter 10 Circles 174
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.5 Ans 3

Question 4.
m\(\widehat{D E F}\)
Big Ideas Math Answer Key Geometry Chapter 10 Circles 175
Answer:
m\(\widehat{D E F}\) = 2(117°) = 234°

Question 5.
m < 1
Big Ideas Math Answer Key Geometry Chapter 10 Circles 176
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.5 Ans 5

Question 6.
m ∠ 3
Big Ideas Math Answer Key Geometry Chapter 10 Circles 177
Answer:
m ∠ 3 = ½ • 140 = 70°

In Exercises 7 – 14, find the value of x.

Question 7.
Big Ideas Math Answer Key Geometry Chapter 10 Circles 178
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.5 Ans 7

Question 8.
Big Ideas Math Answer Key Geometry Chapter 10 Circles 179
Answer:
x° = ½ • (30 + 2x – 30)

Question 9.
Big Ideas Math Answer Key Geometry Chapter 10 Circles 180
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.5 Ans 9

Question 10.
Big Ideas Math Answer Key Geometry Chapter 10 Circles 181
Answer:
34° = ½ (3x – 2 – (x + 6))
34° = ½ (3x – 2 – x – 6)
34° = ½ (2x – 8)
34° = x – 4
x° = 34 + 4
x° = 38°

Question 11.
Big Ideas Math Answer Key Geometry Chapter 10 Circles 182
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.5 Ans 11

Question 12.
Big Ideas Math Answer Key Geometry Chapter 10 Circles 183
Answer:
6x – 11 = 2  • 125
6x = 250 + 11
6x = 261
x° = 43.5°

Question 13.
Big Ideas Math Answer Key Geometry Chapter 10 Circles 184
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.5 Ans 13

Question 14.
Big Ideas Math Answer Key Geometry Chapter 10 Circles 185

Answer:
17x° = 75°
x° = 4.41°

ERROR ANALYSIS
In Exercises 15 and 16, describe and correct the error in finding the angle measure.

Question 15.
Big Ideas Math Answer Key Geometry Chapter 10 Circles 186
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.5 Ans 15

Question 16.
Big Ideas Math Answer Key Geometry Chapter 10 Circles 187
Answer:
m∠1 = ½ (122 – 70)
= ½ (52) = 26
So, m∠1 = 26°

In Exercises 17 – 22, find the indicated angle measure. justify your answer.

Big Ideas Math Answer Key Geometry Chapter 10 Circles 188

Question 17.
m ∠ 1
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.5 Ans 17

Question 18.
m ∠ 2
Answer:
m ∠ 2 = 60°

Explanation:
m ∠ 3 =30°, So, m ∠ 2 = 180° – (90° + 30°)
= 180° – 120° = 60°
Therefore, m ∠ 2 = 60°

Question 19.
m ∠ 3
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.5 Ans 19

Question 20.
m ∠ 4
Answer:
m ∠ 4 = 90°

Question 21.
m ∠ 5
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.5 Ans 21

Question 22.
m ∠ 6
Answer:
m ∠ 6 = 180° – (60° + 30° + 30°) = 180° – 120°
m ∠ 6 = 60°

Question 23.
PROBLEM SOLVING
You are flying in a hot air balloon about 1.2 miles above the ground. Find the measure of the arc that represents the part of Earth you can see. The radius of Earth is about 4000 miles.
Big Ideas Math Answer Key Geometry Chapter 10 Circles 189
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.5 Ans 23

Question 24.
PROBLEM SOLVING
You are watching fireworks over San Diego Bay S as you sail away in a boat. The highest point the fireworks reach F is about 0.2 mile above the bay. Your eyes E are about 0.01 mile above the water. At point B you can no longer see the fireworks because of the curvature of Earth. The radius of Earth is about 4000 miles, and \(\overline{F E}\) is tangent to Earth at point T. Find m\(\widehat{s B}\). Round your answer to the nearest tenth.
Big Ideas Math Answer Key Geometry Chapter 10 Circles 190
Answer:

Question 25.
MATHEMATICAL CONNECTIONS
In the diagram, \(\vec{B}\)A is tangent to ⊙E. Write an algebraic expression for m\(\widehat{C D}\) in terms of x. Then find m\(\widehat{C D}\).
Big Ideas Math Answer Key Geometry Chapter 10 Circles 191
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.5 Ans 25

Question 26.
MATHEMATICAL CONNECTIONS
The circles in the diagram are concentric. Write an algebraic expression for c in terms of a and b.
Big Ideas Math Answer Key Geometry Chapter 10 Circles 192
Answer:
a° = ½(c° – b°)

Question 27.
ABSTRACT REASONING
In the diagram. \(\vec{P}\)L is tangent to the circle, and \(\overline{K J}\) is a diameter. What is the range of possible angle measures of ∠LPJ? Explain your reasoning.
Big Ideas Math Answer Key Geometry Chapter 10 Circles 193
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.5 Ans 27

Question 28.
ABSTRACT REASONING
In the diagram, \(\overline{A B}\) is an chord that is not a diameter of the circle. Line in is tangent to the circle at point A. What is the range of possible values of x? Explain your reasoning. (The diagram is not drawn to scale.)
Big Ideas Math Answer Key Geometry Chapter 10 Circles 194
Answer:
The possible values of x are less than 180°.

Question 29.
PROOF
In the diagram Big Ideas Math Answer Key Geometry Chapter 10 Circles 195 and Big Ideas Math Answer Key Geometry Chapter 10 Circles 196 are secant lines that intersect at point L. Prove that m∠JPN > m∠JLN.
Big Ideas Math Answer Key Geometry Chapter 10 Circles 197
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.5 Ans 29

Question 30.
MAKING AN ARGUMENT
Your friend claims that it is possible for a circumscribed angle to have the same measure as its intercepted arc. Is your friend correct? Explain your reasoning.

Answer:
Yes, when the circumscribed angle measures 90°, the central angle measures 90°, so the intercepted arc also measures 90°.

Question 31.
REASONING
Points A and B are on a circle, and t is a tangent line containing A and another point C.
a. Draw two diagrams that illustrate this situation.
b. Write an equation for m\(\widehat{A B}\) in terms of m∠BAC for each diagram.
c. For what measure of ∠BAC can you use either equation to find m\(\widehat{A B}\)? Explain.
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.5 Ans 31

Question 32.
REASONING
∆XYZ is an equilateral triangle inscribed in ⊙P. AB is tangent to ⊙P at point X, \(\overline{B C}\) is tangent to ⊙P at point Y. and \(\overline{A C}\) is tangent to ⊙P at point Z. Draw a diagram that illustrates this situation. Then classify ∆ABC by its angles and sides. Justify your answer.
Answer:

Question 33.
PROVING A THEOREM
To prove the Tangent and Intersected Chord Theorem (Theorem 10. 14), you must prove three cases.
a. The diagram shows the case where \(\overline{A B}\) contains the center of the circle. Use the Tangent Line to Circle Theorem (Theorem 10.1) to write a paragraph proof for this case.
Big Ideas Math Answer Key Geometry Chapter 10 Circles 198
b. Draw a diagram and write a proof for the case where the center of the circle is in the interior of ∠CAB.
c. Draw a diagram and write a proof for the case where the center of the circle is in the exterior of ∠CAB.
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.5 Ans 33.1
Big Ideas Math Geometry Answers Chapter 10 Circles 10.5 Ans 33.2

Question 34.
HOW DO YOU SEE IT?
In the diagram, television cameras are Positioned at A and B to record what happens on stage. The stage is an arc of ⊙A. You would like the camera at B to have a 30° view of the stage. Should you move the camera closer or farther away? Explain your reasoning.
Big Ideas Math Answer Key Geometry Chapter 10 Circles 199

Answer:
25° = ½(80° – 30°) = ½(50°)
So, you should move the camera closer.

Question 35.
PROVING A THEOREM
Write a proof of the Angles Inside the Circle Theorem (Theorem 10.15).
Big Ideas Math Answer Key Geometry Chapter 10 Circles 200
Given Chords \(\overline{A C}\) and \(\overline{B D}\) intersect inside a circle.
Prove m ∠ 1 = \(\frac{1}{2}\)(m\(\widehat{D C}\) + m\(\widehat{A B}\))
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.5 Ans 35

Question 36.
THOUGHT PROVOKING
In the figure, Big Ideas Math Answer Key Geometry Chapter 10 Circles 201 and Big Ideas Math Answer Key Geometry Chapter 10 Circles 202 are tangent to the circle. Point A is any point on the major are formed by the endpoints of the chord \(\overline{B C}\). Label all congruent angles in the figure. Justify your reasoning.
Big Ideas Math Answer Key Geometry Chapter 10 Circles 203

Answer:
m∠CPB = ½(CAB – CB)

Question 37.
PROVING A THEOREM
Use the diagram below to prove the Angles Outside the Circle Theorem (Theorem 10.16) for the case of a tangent and a secant. Then copy the diagrams for the other two cases on page 563 and draw appropriate auxiliary segments. Use your diagrams to prove each case.
Big Ideas Math Answer Key Geometry Chapter 10 Circles 204
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.5 Ans 37.1
Big Ideas Math Geometry Answers Chapter 10 Circles 10.5 Ans 37.2
Big Ideas Math Geometry Answers Chapter 10 Circles 10.5 Ans 37.3

Question 38.
PROVING A THEOREM
Prove that the Circumscribed Angle Theorem (Theorem 10.17) follows from the Angles Outside the Circle Theorem (Theorem 10.16).
Answer:

In Exercises 39 and 40, find the indicated measure(s). justify your answer

Question 39.
Find m ∠ P when m\(\widehat{W Z Y}\) = 200°
Big Ideas Math Answer Key Geometry Chapter 10 Circles 205
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.5 Ans 39

Question 40.
Find m\(\widehat{A B}\) and m\(\widehat{E D}\)
Big Ideas Math Answer Key Geometry Chapter 10 Circles 206

Answer:
m\(\widehat{E D}\) = ½ (115°) = 57.5°
∠GJA = 30°

Maintaining Mathematical Proficiency

Solve the equation.

Question 41.
x2 + x = 12
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.5 Ans 41

Question 42.
x2 = 12x + 35

Answer:

Explanation:
x² = 12x + 35
x = \(\frac { 12 ± √(144 + 140)  }{ 2 } \)
x = \(\frac { 12 ± √284  }{ 2 } \)
x = \(\frac { 12 + √284  }{ 2 } \), \(\frac { 12 – √284  }{ 2 } \)

Question 43.
– 3 = x2 + 4x
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.5 Ans 43

10.6 Segment Relationships in Circles

Exploration 1

Segments Formed by Two Intersecting Chords

Work with a partner: Use dynamic geometry software.

Sample
Big Ideas Math Geometry Answers Chapter 10 Circles 207

a.
Construct two chords \(\overline{B C}\) and \(\overline{D E}\) that intersect in the interior of a circle at a point F.
Answer:

b.
Find the segment lengths BE, CF, DF, and EF and complete the table. What do you observe?
Big Ideas Math Geometry Solutions Chapter 10 Circles 298
Answer:

c. Repeat parts (a) and (b) several times. Write a conjecture about your results.
REASONING ABSTRACTLY
To be proficient in math, you need to make sense of quantities and their relationships in problem situations.
Answer:

Exploration 2

Secants Intersecting Outside a Circle

Work with a partner: Use dynamic geometry software.

Sample
Big Ideas Math Geometry Answers Chapter 10 Circles 208

a. Construct two secant Big Ideas Math Geometry Answers Chapter 10 Circles 209 and Big Ideas Math Geometry Answers Chapter 10 Circles 210 that intersect at a point B outside a circle, as shown.
Answer:

b. Find the segment lengths BE, BC, BF, and BD. and complete the table. What do you observe?
Big Ideas Math Geometry Solutions Chapter 10 Circles 299
Answer:

c. Repeat parts (a) and (b) several times. Write a conjecture about your results.
Answer:

Communicate Your Answer

Question 3.
What relationships exist among the segments formed by two intersecting chords or among segments of two secants that intersect outside a circle?

Question 4.
Find the segment length AF in the figure at the left.
Big Ideas Math Geometry Answers Chapter 10 Circles 211

Answer:

Explanation:
EA • AF = AD • AC
18 • AF = 9 • 8
AF = 4

Lesson 10.6 Segment Relationships in Circles

Monitoring Progress

Find the value of x.

Question 1.
Big Ideas Math Geometry Answers Chapter 10 Circles 212

Answer:
x = 8

Explanation:
4 • 6 = 3 • x
3x = 24
x = 8

Question 2.
Big Ideas Math Geometry Answers Chapter 10 Circles 213

Answer:
x = 5

Explanation:
2 • x + 1 = 4 • 3
x + 1 = 6
x = 5

Question 3.
Big Ideas Math Geometry Answers Chapter 10 Circles 214

Answer:
x = \(\frac { 54 }{ 5 } \)

Explanation:
6 • 9 = 5 • x
54 = 5x
x = \(\frac { 54 }{ 5 } \)

Question 4.
Big Ideas Math Geometry Answers Chapter 10 Circles 215

Answer:
x = \(\frac { 3 ± √37 }{ 2 } \)

Explanation:
3 • x + 2 = x + 1 • x – 1
3x + 6 = x² – 1
x² – 3x – 7 = 0
x = \(\frac { 3 ± √(9 + 28) }{ 2 } \)
x = \(\frac { 3 ± √37 }{ 2 } \)

Question 5.
Big Ideas Math Geometry Answers Chapter 10 Circles 216

Answer:
x = ±√3

Explanation:
x² = 3 • 1
x² = 3
x = ±√3

Question 6.
Big Ideas Math Geometry Answers Chapter 10 Circles 217

Answer:
x = \(\frac { 49 }{ 5 } \)

Explanation:
7² = 5 • x
49 = 5x
x = \(\frac { 49 }{ 5 } \)

Question 7.
Big Ideas Math Geometry Answers Chapter 10 Circles 218

Answer:
x = 14.4

Explanation:
12² = 10x
144 = 10x
x = 14.4

Question 8.
WHAT IF?
In Example 4, CB = 35 feet and CE = 14 feet. Find the radius of the tank.

Answer:
The radius of the tank is 36.75

Explanation:
CB² = CE ⋅ CD
35² = 14 ⋅ (2r + 14)
1225 = 28r + 196
28r = 1029
r = 36.75

Exercise 10.6 Segment Relationships in Circles

Vocabulary and Core Concept Check

Question 1.
VOCABULARY
The part of the secant segment that is outside the circle is called a(n) ______________ .
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.6 Ans 1

Question 2.
WRITING
Explain the difference between a tangent segment and a secant segment.

Answer:
A tangent segment intersects the circle at only one point. It actually doesn’t go through the circle. If a ball is rolling on a table top, then it would be the tangent. A secant segment intersects the circle in two points. It goes through the circle. In a tangent, no part is in the interior of the circle. In a secant, there is a part in the interior called a chord.

Monitoring Progress and Modeling with Mathematics

In Exercises 3 – 6, find the value of x.

Question 3.
Big Ideas Math Geometry Answers Chapter 10 Circles 219
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.6 Ans 3

Question 4.
Big Ideas Math Geometry Answers Chapter 10 Circles 220

Answer:
x = 23

Explanation:
10 • 18 = 9 • (x – 3)
20 = x – 3
x = 23

Question 5.
Big Ideas Math Geometry Answers Chapter 10 Circles 221
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.6 Ans 5

Question 6.
Big Ideas Math Geometry Answers Chapter 10 Circles 222

Answer:
x = 5

Explanation:
2x • 12 = 15 • (x + 3)
24x = 15x + 45
9x = 45
x = 5

In Exercises 7 – 10, find the value of x.

Question 7.
Big Ideas Math Geometry Answers Chapter 10 Circles 223
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.6 Ans 7

Question 8.
Big Ideas Math Geometry Answers Chapter 10 Circles 224

Answer:
x = \(\frac { 35 }{ 4 } \)

Explanation:
5 • 7 = 4 • x
4x = 35
x = \(\frac { 35 }{ 4 } \)

Question 9.
Big Ideas Math Geometry Answers Chapter 10 Circles 225
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.6 Ans 9

Question 10.
Big Ideas Math Geometry Answers Chapter 10 Circles 226

Answer:
x = 30

Explanation:
45 • x = 50 • 27
45x = 1350
x = 30

In Exercises 11 – 14. find the value of x.

Question 11.
Big Ideas Math Geometry Answers Chapter 10 Circles 227
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.6 Ans 11

Question 12.
Big Ideas Math Geometry Answers Chapter 10 Circles 228

Answer:
x = 48

Explanation:
24² = 12x
576 = 12x
x = 48

Question 13.
Big Ideas Math Geometry Answers Chapter 10 Circles 229
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.6 Ans 13

Question 14.
Big Ideas Math Geometry Answers Chapter 10 Circles 230

Answer:
x = 1.5

Explanation:
3 = 2x
x = 1.5

Question 15.
ERROR ANALYSIS
Describe and correct the error in finding CD.
Big Ideas Math Geometry Answers Chapter 10 Circles 231
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.6 Ans 15

Question 16.
MODELING WITH MATHEMATICS
The Cassini spacecraft is on a mission in orbit around Saturn until September 2017. Three of Saturn’s moons. Tethys. Calypso, and Teleslo. have nearly circular orbits of radius 2,95,000 kilometers. The diagram shows the positions of the moons and the spacecraft on one of Cassini’s missions. Find the distance DB from Cassini to Tethys when \(\overline{A D}\) is tangent to the circular orbit.
Big Ideas Math Geometry Answers Chapter 10 Circles 232

Answer:
BD = 579493 km

Explanation:
(203,000)² = 83000x
x = 496493
BC = 496493
BD = 496493 + 83000 = 579493

Question 17.
MODELING WITH MATHEMATICS
The circular stone mound in Ireland called Newgrange has a diameter of 250 feet. A passage 62 feet long leads toward the center of the mound. Find the perpendicular distance x from the end of the passage to either side of the mound.
Big Ideas Math Geometry Answers Chapter 10 Circles 233
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.6 Ans 17

Question 18.
MODELING WITH MATHEMATICS
You are designing an animated logo for our website. Sparkles leave point C and move to the Outer circle along the segments shown so that all of the sparkles reach the outer circle at the same time. Sparkles travel from point C to point D at 2 centimeters per second. How fast should sparkles move from point C to point N? Explain.
Big Ideas Math Geometry Answers Chapter 10 Circles 234

Answer:
5.33 should sparkles move from point C to point N.

Explanation:
4 • 8 = 6 • x
x = 5.33

Question 19.
PROVING A THEOREM
Write a two-column proof of the Segments of Chords Theorem (Theorem 10.18).

Plan for Proof:
Use the diagram to draw \(\overline{A C}\) and \(\overline{D B}\). Show that ∆EAC and ∆EDB are similar. Use the fact that corresponding side lengths in similar triangles are proportional.
Big Ideas Math Geometry Answers Ch
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.6 Ans 19.1
Big Ideas Math Geometry Answers Chapter 10 Circles 10.6 Ans 19.2

Question 20.
PROVING A THEOREM
Prove the Segments of Secants Theorem (Theorem 10.19). (Hint: Draw a diagram and add auxiliary line segments to form similar triangles.)
Answer:

Question 21.
PROVING A THEOREM
Use the Tangent Line to Circle Theorem (Theorem 10. 1) to prove the Segments of Secants and Tangents Theorem (Theorem 10.20) for the special case when the secant segment Contains the center of the circle.
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.6 Ans 21

Question 22.
PROVING A THEOREM
Prove the Segments of Secants and Tangents Theorem (Theorem 10.20). (Hint: Draw a diagram and add auxiliary line segments to form similar triangles.)
Answer:

Question 23.
WRITING EQUATIONS
In the diagram of the water well, AB, AD, and DE are known. Write an equation for BC using these three measurements.
Big Ideas Math Geometry Answers Chapter 10 Circles 236
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.6 Ans 23

Question 24.
HOW DO YOU SEE IT?
Which two theorems would you need to use to tind PQ? Explain your reasoning.
Big Ideas Math Geometry Answers Chapter 10 Circles 237

Answer:

Question 25.
CRITICAL THINKING
In the figure, AB = 12, BC = 8, DE = 6, PD = 4, and a is a point of tangency. Find the radius of ⊙P.
Big Ideas Math Geometry Answers Chapter 10 Circles 238
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.6 Ans 25

Question 26.
THOUGHT PROVOKING
Circumscribe a triangle about a circle. Then, using the points of tangency, inscribe a triangle in the circle. Must it be true that the two triangles are similar? Explain your reasoning.

Answer:

Maintaining Mathematical Proficiency

Solve the equation by completing the square.

Question 27.
x2 + 4x = 45
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.6 Ans 27

Question 28.
x2 – 2x – 1 = 8

Answer:
x = 1 + √10, x = 1 – √10

Explanation:
x² – 2x – 1 = 8
x² – 2x – 9 = 0
x = \(\frac { 2 ± √(4 + 36) }{ 2 } \)
x = \(\frac { 2 ± √40 }{ 2 } \)
x = 1 + √10, x = 1 – √10

Question 29.
2x2 + 12x + 20 = 34
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.6 Ans 29

Question 30.
– 4x2 + 8x + 44 = 16

Answer:
x = 1 + √8, x = 1 – √8

Explanation:
– 4x² + 8x + 44 = 16
4x² – 8x – 28 =0
x² – 2x – 7 = 0
x = \(\frac { 2 ± √(4 + 28) }{ 2 } \)
x = \(\frac { 2 ± √32 }{ 2 } \)
x = 1 ± √8

10.7 Circles in the Coordinate Plane

Exploration 1

The Equation of a Circle with Center at the Origin

Work with a partner: Use dynamic geometry software to Construct and determine the equations of circles centered at (0, 0) in the coordinate plane, as described below.
Big Ideas Math Geometry Solutions Chapter 10 Circles 300
a. Complete the first two rows of the table for circles with the given radii. Complete the other rows for circles with radii of your choice.

Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 18

b. Write an equation of a circle with center (0, 0) and radius r.

Answer:
x²  + y²  = r²

Explanation:
(x – 0)² + (y – 0)²  = r²
x²  + y²  = r²

Exploration 2

The Equation of a Circle with Center (h, k)

Work with a partner: Use dynamic geometry software to construct and determine the equations of circles of radius 2 in the coordinate plane, as described below.
Big Ideas Math Geometry Solutions Chapter 10 Circles 301
a. Complete the first two rows of the table for circles with the given centers. Complete the other rows for circles with centers of your choice.

Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 17

b. Write an equation of a circle with center (h, k) and radius 2.

Answer:
(x – h)² + (y – k)² = 4

c. Write an equation of a circle with center (h, k) and radius r.

Answer:
(x – h)² + (y – k)² = r²

Exploration 3

Deriving the Standard Equation of a Circle

Work with a partner. Consider a circle with radius r and center (h, k).

Write the Distance Formula to represent the distance d between a point (x, y) on the circle and the center (h, k) of the circle. Then square each side of the Distance Formula equation.

How does your result compare with the equation you wrote in part (c) of Exploration 2?

MAKING SENSE OF PROBLEMS
To be proficient in math, you need to explain correspondences between equations and graphs.
Big Ideas Math Answers Geometry Chapter 10 Circles 239

Answer:
(x – h)² + (y – k)² = r²

Communicate Your Answer

Question 4.
What is the equation of a circle with center (h, k) and radius r in the coordinate plane?

Answer:
(x – h)² + (y – k)² = r²

Question 5.
Write an equation of the circle with center (4, – 1) and radius 3.

Answer:
x² + y² – 8x + 2y + 8 = 0

Explanation:
(x – 4)² + (y + 1)² = 9
x² – 8x + 16 + y² + 2y + 1 = 9
x² + y² – 8x + 2y = 9 – 17
x² + y² – 8x + 2y + 8 = 0

Lesson 10.7 Circles in the Coordinate Plane

Monitoring Progress

Write the standard equation of the circle with the given center and radius.

Question 1.
center: (0, 0), radius: 2.5

Answer:
x² + y² = 6.25

Explanation:
(x – 0)² + (y – 0)² = 2.5²
x² + y² = 6.25

Question 2.
center: (- 2, 5), radius: 7

Answer:
(x + 2)² + (y – 5)² = 49

Explanation:
(x + 2)² + (y – 5)² = 7²
(x + 2)² + (y – 5)² = 49

Question 3.
The point (3, 4) is on a circle with center (1, 4). Write the standard equation of the circle.

Answer:
(x – 1)² + (y – 4)² = 4

Explanation:
r = √(3 – 1)² + (4 – 4)²
= √(2)²
r = 2
(x – 1)² + (y – 4)² = 2²
(x – 1)² + (y – 4)² = 4

Question 4.
The equation of a circle is x2 + y2 – 8x + 6y + 9 = 0. Find the center and the radius of the circle. Then graph the circle.

Answer:
The center of the circle (4, -3) and radius is 4.

Explanation:
x² + y² – 8x + 6y + 9 = 0
x² – 8x + 16 + y² + 6y + 9 = 16
(x – 4)² + (y + 3)² = 4²
The center of the circle (4, -3) and radius is 4.

Question 5.
Prove or disprove that the point (1, √5 ) lies on the circle centered at the origin and containing the point (0, 1).

Answer:
Disproved.

Explanation:
We consider the circle centred at the origin and containing the point (0, 1).
Therefore, we can conclude that the radius of the circle r = 1, let O (0, 0) and B (1, √5). So the distance between two points is
OB = √(1 – 0) + (√5 – 0)² = √(1 + 5) = √6
As the radius of the given circle is 1 and distance of the point B from its centre is √6. So we can conclude that point does lie on the given circle.

Question 6.
why are three seismographs needed to locate an earthquake’s epicentre?

Answer:

Exercise 10.7 Circles in the Coordinate Plane

Vocabulary and Core Concept Check

Question 1.
VOCABULARY
What is the standard equation of a circle?
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.7 Ans 1

Question 2.
WRITING
Explain why knowing the location of the center and one point on a circle is enough to graph the circle.

Answer:
If we know the location of the center and one point on the circle, we can graph a circle because the distance from the center to the point is called the radius.

Monitoring Progress and Modeling with Mathematics

In Exercises 3 – 8, write the standard equation of the circle.

Question 3.
Big Ideas Math Answers Geometry Chapter 10 Circles 240
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.7 Ans 3

Question 4.
Big Ideas Math Answers Geometry Chapter 10 Circles 241

Answer:
x² + y² = 36

Explanation:
The center is (0, 0) and the radius is 6
(x – h)² + (y – k)² = r²
(x – 0)² + (y – 0)² = 6²
x² + y² = 36

Question 5.
a circle with center (0, 0) and radius 7
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.7 Ans 5

Question 6.
a circle with center (4, 1) and radius 5

Answer:
(x – 4)² + (y – 1)² = 25

Explanation:
(x – h)² + (y – k)² = r²
(x – 4)² + (y – 1)² = 5²
(x – 4)² + (y – 1)² = 25

Question 7.
a circle with center (- 3, 4) and radius 1
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.7 Ans 7

Question 8.
a circle with center (3, – 5) and radius 7

Answer:
(x – 3)² + (y + 5)² = 49

Explanation:
(x – h)² + (y – k)² = r²
(x – 3)² + (y + 5)² = 7²
(x – 3)² + (y + 5)² = 49

In Exercises 9 – 11, use the given information to write the standard equation of the circle.

Question 9.
The center is (0, 0), and a point on the circle is (0, 6).
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.7 Ans 9

Question 10.
The center is (1, 2), and a point on the circle is (4, 2).

Answer:
x² + y² = 9

Explanation:
r = √(x – h)² + (y – k)²
= √(4 – 1)² + (2 – 2)²
= √3²
r = 3
(x – h)² + (y – k)² = r²
(x – 0)² + (y – 0)² = 3²
x² + y² = 9

Question 11.
The center is (0, 0). and a point on the circle is (3, – 7).
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.7 Ans 11

Question 12.
ERROR ANALYSIS
Describe and correct the error in writing the standard equation of a circle.
Big Ideas Math Answers Geometry Chapter 10 Circles 242

Answer:
(x – h)² + (y – k)² = r²
(x + 3)² + (y + 5)² = 3²
(x + 3)² + (y + 5)² = 9

In Exercises 13 – 18, find the center and radius of the circle. Then graph the circle.

Question 13.
x2 + y2 = 49
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.7 Ans 13

Question 14.
(x + 5)2 + (y – 3)2 = 9

Answer:
Center is (-5, 3) and rdaius is 3.

Explanation:
For the equation (x + 5)2 + (y – 3)2 = 9, center is (-5, 3) and rdaius is 3.

Question 15.
x2 + y2 – 6x = 7
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.7 Ans 15

Question 16.
x2 + y2 + 4y = 32

Answer:
The center is (0, -2) and radius is 6

Explanation:
x2 + y2 + 4y = 32
x² + y² + 4y + 4 = 32 + 4
x² + (y + 2)² = 36
(x – 0)² + (y – (-2))² = 6²
The center is (0, -2) and radius is 6
Big Ideas Math Geometry Answers Chapter 10 Circles 21

Question 17.
x2 + y2 – 8x – 2y = – 16
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.7 Ans 17

Question 18.
x2 + y2 + 4x + 12y = – 15

Answer:
The center is (-2, -6) and radius is 5

Explanation:
x2 + y2 + 4x + 12y = – 15
x² + 4x + 4 + y² + 12y + 36 = -15 + 36 + 4
(x + 2)² + (y + 6)² = 5²
The center is (-2, -6) and radius is 5

In Exercises 19 – 22, prove or disprove the statement.

Question 19.
The point (2, 3) lies on the circle centered at the origin with radius 8.
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.7 Ans 19

Question 20.
The point (4, √5) lies on the circle centered at the origin with radius 3.

Answer:
The point (4, √5) does not lie on the circle.

Explanation:
r² = (x – h)² + (y – k)²
3² ______________ (4 – 0)² + (√5 – 0)²
9 ______________ 16 + 5
9 ≠ 21
Because the radius is 3 and the distance between center and the point is more than the radius. So the point does not lie on the circle.

Question 21.
The point (√6, 2) lies on the circle centered at the origin and containing the point (3, – 1).
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.7 Ans 21

Question 22.
The point (√7, 5) lies on the circle centered at the origin and containing the point (5, 2).

Answer:
The point (√7, 5) does not lie on the circle.

Explanation:
r² = (x – h)² + (y – k)²
= (√7 – 0)² + (5 – 0)² = 7 + 25 = 32
r = 5.65
(5.65)² ______________ (5 – 0)² + (2 – 0)²
32 ______________ 25 + 4
32 ≠ 29
Because the radius is 3 and the distance between center and the point is more than the radius. So the point does not lie on the circle.

Question 23.
MODELING WITH MATHEMATICS
A City’s commuter system has three zones. Zone I serves people living within 3 miles of the city’s center. Zone 2 serves those between 3 and 7 miles from the center. Zone 3 serves those over 7 miles from the center.
Big Ideas Math Answers Geometry Chapter 10 Circles 243
a. Graph this Situation on a coordinate plane where each unit corresponds to 1 mile. Locate the city’s center at the origin.
b. Determine which zone serves people whose homes are represented by the points (3, 4), (6, 5), (1, 2), (0.3). and (1, 6).
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.7 Ans 23.1
Big Ideas Math Geometry Answers Chapter 10 Circles 10.7 Ans 23.2

Question 24.
MODELING WITH MATHEMATICS
Telecommunication towers can be used to transmit cellular phone calls. A graph with units measured in kilometers shows towers at points (0, 0), (0, 5), and (6, 3). These towers have a range of about 3 kilometers.
a. Sketch a graph and locate the towers. Are there any locations that may receive calls from more than one tower? Explain your reasoning.
Answer:

b. The center of City A is located at (- 2, 2.5), and the center of City B is located at (5, 4). Each city has a radius of 1.5 kilometers. Which city seems to have better cell phone coverage? Explain your reasoning.

Answer:
There are three towers at points (0, 0), (0, 5) and (6, 3) with range of about 3 km
a. Let’s sketch the graph to locate towers. Draw the points (0, 0), (0, 5) and (6, 3). Then draw three circles with centers at these points and radii 3.
There are locations that can receive calls from more that one tower because circles with centers (0, 0) and (0, 5) overlap. Locations in their intersection can receive calls from two towers.
The city A has a center located at (-2, 2.5) and city B has a center located at (5, 4). Both cities have radius 1.5 km
Let’s draw the city A as a circle with center (-2, 2.5) and radius 1.5 and city B with center (5, 4) and radius 1.5.
From the graph we can conclude that the city B has better cell phone coverage because parts of city A do not have coverage.

Question 25.
REASONING
Sketch the graph of the circle whose equation is x2 + y2 = 16. Then sketch the graph of the circle after the translation (x, y) → (x – 2, y – 4). What is the equation of the image? Make a conjecture about the equation of the image of a circle centered at the origin after a translation m units to the left and n units down.
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.7 Ans 25

Question 26.
HOW DO YOU SEE IT?
Match each graph with its equation.

a. Big Ideas Math Answers Geometry Chapter 10 Circles 244 A. x2 + (y + 3) 2 = 4
b. Big Ideas Math Answers Geometry Chapter 10 Circles 245 B. (x – 3) 2 + y2 = 4
c. Big Ideas Math Answers Geometry Chapter 10 Circles 246 C. (x + 3) 2 + y2 = 4
d. Big Ideas Math Answers Geometry Chapter 10 Circles 247 D. x2 + (y – 3) 2 = 4

Answer:
a ➝ C, b ➝ A, c ➝ D, d ➝ B

Question 27.
USING STRUCTURE
The vertices of ∆XYZ are X(4, 5), Y(4, 13), and Z(8, 9). Find the equation of the circle circumscribed about ∆XYZ. Justify your answer.
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.7 Ans 27.1
Big Ideas Math Geometry Answers Chapter 10 Circles 10.7 Ans 27.2

Question 28.
THOUGHT PROVOKING
A circle has center (h, k) and contains point (a, b). Write the equation of the line tangent to the circle at point (a, b).

Answer:
y – b = \(\frac { h – a }{ b – k } \)(x – a)

Explanation:
It is given a circle with center C(h, k. A circle with point A(a, b). We have to write the equation of a tangent that intersects the circle at point A
By the tangent line to circle theorem, a tangent is perpendicular to the radius. Two lines are perpendicular if and only if their slopes are negative reciprocals. So, find the equation of the line AC to know its slope.
The equation of the line which has two points (a, b), (c, d) is y – b = \(\frac { b – c }{ a – d } \)(x – a)
The equation of the line which has two points A(a, b) and C(h, k) is
y – b = \(\frac { b – k }{ a – h } \)(x – a)
Therefore, the slope of the line through A nad C is \(\frac { b – k }{ a – h } \)
Hence the slope of the tangent is –\(\frac { a – h }{ b – k } \) = \(\frac { h – a }{ b – k } \)
Use the equation of the line y = kx + n through the point (a, b)
y – b = k(x – a) to find the equation of the tangent
The equation of the tangent with slope \(\frac { h – a }{ b – k } \) and through the point A(a, b) is y – b = \(\frac { h – a }{ b – k } \)(x – a)

MATHEMATICAL CONNECTIONS
In Exercises 29 – 32, use the equations to determine whether the line is a tangent, a secant a secant that contains the diameter, or name of these. Explain your reasoning.

Question 29.
Circle: (x – 4)2 + (y – 3)2 = 9
Line: y = 6
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.7 Ans 29.1

Question 30.
Circle: (x + 2)2 + (y – 2)2 = 16
Line: y = 2x – 4

Answer:
The line is a secant line.

Explanation:
(x + 2)2 + (y – 2)2 = 16, y = 2x – 4
(x + 2)2 + (2x – 4 – 2)2 = 16
x² + 4x + 4 + (2x – 6)² = 16
x² + 4x + 4 + 4x² – 24x + 36 = 16
5x² – 20x + 40 – 16 = 0
5x² – 20x + 24 = 0
x = \(\frac { 20 ±√-80 }{ 10 } \)
x = 2, y = 2 • 2 – 4 = 0, (2, 0)
The system has two solutions and point does not lie on the line. So the line is a secant line.

Question 31.
Circle: (x – 5)2 + (y + 1)2 = 4
Line: y = \(\frac{1}{5}\)x – 3\
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.7 Ans 31.1
Big Ideas Math Geometry Answers Chapter 10 Circles 10.7 Ans 31.2
Big Ideas Math Geometry Answers Chapter 10 Circles 10.7 Ans 31.3

Question 32.
Circle: (x + 3)2 + (y – 6)2 = 25
Line: y = –\(\frac{4}{3}\)x + 2

Answer:
The line is a secant line.

Explanation:
(x + 3)2 + (y – 6)2 = 25, y = –\(\frac{4}{3}\)x + 2
(x + 3)2 + (-\(\frac{4}{3}\)x + 2 – 6)2 = 25
x² + 6x + 9 + \(\frac { 16x² }{ 9 } \) + \(\frac { 32x }{ 3 } \) + 16 = 25
\(\frac { 25x² }{ 9 } \) + \(\frac { 50x }{ 3 } \) = 0
x(25x + 150) = 0
x = 0 or x = -6
y = 2, y = 10
(0, 2) and (-6, 10)
d = √(0 + 6)² + (2 – 10)²
= √(36 + 64)
= 10 ≠ 5
The system has two solutions and point does not lie on the line. So the line is a secant line.

MAKING AN ARGUMENT
Question 33.
Your friend claims that the equation of a circle passing through the points (- 1, 0) and (1, 0) is x2 – 2yk + y2 = 1 with center (0, k). Is your friend correct? Explain your reasoning.
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.7 Ans 33

Question 34.
REASONING
Four tangent circles are centered on the x-axis. The radius of ⊙A is twice the radius of ⊙O, The radius of ⊙B is three times the radius of ⊙O, The radius of ⊙C is four times the radius of ⊙O, All circles have integer radii, and the point (63, 16) is On ⊙C. What is the equation of ⊙A? Explain your reasoning.
Big Ideas Math Answers Geometry Chapter 10 Circles 248
Answer:

Maintaining Mathematical Proficiency

Identify the arc as a major arc, minor arc, or semicircle. Then find the measure of the arc.

Big Ideas Math Answers Geometry Chapter 10 Circles 249

Question 35.
\(\widehat{R S}\)
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.7 Ans 35

Question 36.
\(\widehat{P R}\)

Answer:
\(\widehat{P R}\) is a right angle
\(\widehat{P R}\) = 25 + 65 = 90°

Question 37.
\(\widehat{P R T}\)
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.7 Ans 37

Question 38.
\(\widehat{S T}\)

Answer:
\(\widehat{S T}\) is a major arc
\(\widehat{S T}\) = 360 – (90 + 65 +25 + 53) = 127°

Question 39.
\(\widehat{R S T}\)
Answer:
Big Ideas Math Geometry Answers Chapter 10 Circles 10.7 Ans 39

Question 40.
\(\widehat{Q S}\)

Answer:
\(\widehat{Q S}\) is a minor arc
\(\widehat{Q S}\) = 25 + 53 = 78°

Circles Review

10.1 Lines and Segments That Intersect Circles

Tell whether the line, ray, or segment is best described as a radius, chord, diameter, secant, or tangent of ⊙P.

Big Ideas Math Geometry Answer Key Chapter 10 Circles 250

Question 1.
\(\overline{P K}\)

Answer:
\(\overline{P K}\) is radius

Question 2.
\(\overline{N M}\)

Answer:
\(\overline{N M}\) is chord

Question 3.
\(\vec{J}\)L

Answer:
\(\vec{J}\)L is tangent

Question 4.
\(\overline{K N}\)

Answer:
\(\overline{K N}\) is diameter

Question 5.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 251

Answer:
NL is secant

Question 6.
\(\overline{P N}\)

Answer:
\(\overline{P N}\) is radius

Tell whether the common tangent is internal or external.

Question 7.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 252

Answer:
Internal common tangent

Question 8.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 253

Answer:
External common tangent

Points Y and Z are points of tangency. Find the value of the variable.

Question 9.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 254

Answer:
a = \(\frac { 3 ± 33 }{ 18 } \)

Explanation:
3a = 9a² – 30
9a² – 3a – 30 = 0
a = \(\frac { 3 ± 33 }{ 18 } \)

Question 10.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 255

Answer:
c = 2

Explanation:
2c² + 9c + 6 = 9c + 14
2c² – 8 = 0
c² – 4 = 0
c = 2

Question 11.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 256

Answer:
r = 12

Explanation:
(3 + r)² = r² + 9²
9 + 6r + r² = r² + 81
6r = 72
r = 12

Question 12.
Tell whether \(\overline{B D}\) is tangent to ⊙C. Explain.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 257

Answer:
\(\overline{B D}\) is not tangent to ⊙C

Explanation:
52² = 10² + 48²
2704 = 100 + 2304
So, \(\overline{B D}\) is not tangent to ⊙C

10.2 Finding Arc Measures

Use the diagram above to find the measure of the indicated arc.

Question 13.
\(\widehat{K L}\)

Answer:
\(\widehat{K L}\) = 100°

Explanation:
\(\widehat{K L}\) = ∠KPL = 100°

Question 14.
\(\widehat{L M}\)

Answer:
\(\widehat{L M}\) = 60°

Explanation:
\(\widehat{L M}\) = 180° – 120°
= 60°

Question 15.
\(\widehat{K M}\)

Answer:
\(\widehat{K M}\) = 160°

Explanation:
\(\widehat{K M}\) = 100° + 60°
= 160°

Question 16.
\(\widehat{K N}\)

Answer:
\(\widehat{K N}\) = 80°

Explanation:
\(\widehat{K N}\) = 360 – (120 + 100 + 60)
= 360 – 280 = 80°

Tell whether the red arcs are congruent. Explain why or why not.

Question 17.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 258

Answer:
\(\widehat{S T}\), \(\widehat{Y Z}\) are not congruent.

Explanation:
\(\widehat{S T}\), \(\widehat{Y Z}\) are not congruent. Because both arcs are from different circles and having different radii.

Question 18.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 259

Answer:
\(\widehat{A B}\), \(\widehat{E F}\) are congruent.

Explanation:
\(\widehat{A B}\), \(\widehat{E F}\) are congruent. Because those circles have same radii.

10.3 Using Chords

Find the measure of \(\widehat{A B}\).

Question 19.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 260

Answer:
\(\widehat{A B}\) = 61°

Explanation:
\(\widehat{A B}\) = 61°
If ED = AB, then \(\widehat{A B}\) = \(\widehat{E D}\)

Question 20.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 261

Answer:
\(\widehat{A B}\) = 65°

Explanation:
\(\widehat{A B}\) = \(\widehat{A D}\)
So, \(\widehat{A B}\) = 65°

Question 21.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 262

Answer:
\(\widehat{A B}\) = 91°

Explanation:
\(\widehat{A B}\) = \(\widehat{E D}\)
So, \(\widehat{A B}\) = 91°

Question 22.
In the diagram. QN = QP = 10, JK = 4x, and LM = 6x – 24. Find the radius of ⊙Q.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 263

Answer:
The radius of ⊙Q is 26.

Explanation:
6x – 24 = 4x
6x – 4x = 24
2x = 24
x = 12
ML = 6(12) – 24 = 48
JN = \(\frac { 48 }{ 2 } \) = 24
JQ² = JN² + NQ²
= 24² + 10² = 576 + 100
JQ = 26
Therefore, the radius of ⊙Q is 26

10.4 Inscribed Angles and Polygons

Find the value(s) of the variable(s).

Question 23.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 264

Answer:
x° = 80°

Explanation:
x° = 2 • 40° = 80°

Question 24.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 265

Answer:
q° = 100°, r° = 20°

Explanation:
q° + 80° = 180°
q° = 100°
4r° + 100 = 180°
4r° = 80°
r° = 20°

Question 25.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 266

Answer:
d° = 5°

Explanation:
14d° = 70°
d° = 5°

Question 26.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 267

Answer:
y° = 30°, z° = 10°

Explanation:
3y° = 90°
y° = 30°
50° + 90° + 4z° = 180°
4z° = 40°
z° = 10°

Question 27.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 268

Answer:
m° = 44°
n° = 39°

Explanation:
m° = 44°
n° = 39°

Question 28.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 269

Answer:
c° = 28°

Explanation:
c° = ½ • 56 = 28

10.5 Angle Relationships in Circles

Find the value of x.

Question 29.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 270

Answer:
x° = 250°

Explanation:
x° = 250°

Question 30.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 271

Answer:
x° = 106°

Explanation:
x° = ½(152 + 60)
= ½(212) = 106°

Question 31.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 272

Answer:
x° = 28°

Explanation:
x° = ½(96 – 40)
= ½(56) = 28°

Question 32.
Line l is tangent to the circle. Find m\(\widehat{X Y Z}\).
Big Ideas Math Geometry Answer Key Chapter 10 Circles 273

Answer:
m\(\widehat{X Y Z}\) = 240°

Explanation:
m\(\widehat{X Y Z}\) = 2(120)
= 240°

10.6 Segment Relationships in Circles

Find the value of x.

Question 33.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 274

Answer:
x = 8

Explanation:
3 • x = 4 • 6
x = 8

Question 34.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 275

Answer:
x = 3

Explanation:
(x + 3) • x = (6 – x) • 2x
x + 3 = 12 – 2x
3x = 9
x = 3

Question 35.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 276

Answer:
x = 18

Explanation:
12² = 8 • x
144 = 8x
x = 18

Question 36.
A local park has a circular ice skating rink. You are standing at point A, about 12 feet from the edge of the rink. The distance from you to a point of tangency on the rink is about 20 feet. Estimate the radius of the rink.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 277

Answer:
Estimated radius of the rink is 10 ft

Explanation:
20² = 12 • (2r + 12)
400 = 24r + 144
256 = 24r
r = 10.66
Therefore, estimated radius of the rink is 10 ft

10.7 Circles in the Coordinate Plane

Write the standard equation of the circle shown.

Question 37.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 278

Answer:
(x – 4)² + (y + 1)² = 12.25

Explanation:
(x – 4)² + (y + 1)² = 3.5²
(x – 4)² + (y + 1)² = 12.25

Question 38.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 279

Answer:
(x – 8)² + (y – 5)² = 36

Explanation:
(x – 8)² + (y – 5)² = 6²
(x – 8)² + (y – 5)² = 36

Question 39.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 280

Answer:
x² + y² = 4

Explanation:
(x – 0)² + (y – 0)² = 2²
x² + y² = 4

Write the standard equation of the circle with the given center and radius.

Question 40.
center: (0,0), radius: 9

Answer:
x² + y² = 81

Explanation:
(x – 0)² + (y – 0)² = 9²
x² + y² = 81

Question 41.
center: (- 5, 2), radius: 1.3

Answer:
(x + 5)² + (y – 2)² = 1.69

Explanation:
(x + 5)² + (y – 2)² = 1.3²
(x + 5)² + (y – 2)² = 1.69

Question 42.
center: (6, 21), radius: 4

Answer:
(x – 6)² + (y – 21)² = 16

Explanation:
(x – 6)² + (y – 21)² = 4²
(x – 6)² + (y – 21)² = 16

Question 43.
center: (- 3, 2), radius: 16

Answer:
(x + 3)² + (y – 2)² = 256

Explanation:
(x + 3)² + (y – 2)² = 16²
(x + 3)² + (y – 2)² = 256

Question 44.
center: (10, 7), radius: 3.5

Answer:
(x – 10)² + (y – 7)² = 12.25

Explanation:
(x – 10)² + (y – 7)² = 3.5²
(x – 10)² + (y – 7)² = 12.25

Question 45.
center: (0, 0), radius: 5.2

Answer:
x² + y² = 27.04

Explanation:
(x – 0)² + (y – 0)² = 5.2²
x² + y² = 27.04

Question 46.
The point (- 7, 1) is on a circle with center (- 7, 6). Write the standard equation of the circle.

Answer:
(x + 7)² + (y – 6)² = 25

Explanation:
r² = (-7 + 7)² + (6 – 1)²
= 5²
r = 5
And, centre is (-7, 6)
The standard equation of a circle is (x – (-7))² + (y – 6)² = 5²
(x + 7)² + (y – 6)² = 25

Question 47.
The equation of a circle is x2 + y2 – 12x + 8y + 48 = 0. Find the center and the radius of the circle. Then graph the circle.

Answer:
The radius of the circle is 2, the centre is (6, -4)

Explanation:
x2 + y2 – 12x + 8y + 48 = 0
x² – 12x + 36 + y² + 8y + 16 = 4
(x – 6)² + (y + 4)² = 2²
So, the radius of the circle is 2, the centre is (6, -4)
Big Ideas Math Geometry Answers Chapter 10 Circles 23

Question 48.
Prove or disprove that the point (4, – 3) lies on the circle centred at the origin and containing
the point (- 5, 0).

Answer:
The point (4, – 3) lies on the circle.

Explanation:
Use the distance formula to find the radius of the circle with cente (0, 0) and a point (-5, 0)
r = √(-5 – 0)² + (0 – 0)² = 5
The distance from the point (4, -3) to the center (0, 0)
d = √(4 – 0)² + (-3 – 0)² = √(16 +9) = 5
Since the radius of the circle is 5, the point lies on the circle.

Circles Chapter Test

Find the measure of each numbered angle in ⊙P. Justify your answer.

Question 1.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 281

Answer:
m∠1 = 72.5°
m∠2 = 145°

Explanation:
m∠1 = \(\frac { 145 }{ 2 } \)
= 72.5°
m∠2 = 145°

Question 2.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 282

Answer:
m∠1 = 60°, m∠2 = 90°

Explanation:
A tangent is perpendicualr to diameter. So, m∠2 = 90°
m∠1 = 60°

Question 3.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 283

Answer:
m∠1 = 48°

Explanation:
m∠1 = \(\frac { 96 }{ 2 } \) = 48°

Question 4.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 284

Answer:

Use the diagram.

Big Ideas Math Geometry Answer Key Chapter 10 Circles 285

Question 5.
AG = 2, GD = 9, and BG = 3. Find GF.

Answer:

Question 6.
CF = 12, CB = 3, and CD = 9. Find CE.

Answer:

Question 7.
BF = 9 and CB = 3. Find CA

Answer:

Question 8.
Sketch a pentagon inscribed in a circle. Label the pentagon ABCDE. Describe the relationship between each pair of angles. Explain your reasoning.

a. ∠CDE and ∠CAE

Answer:

b. ∠CBE and ∠CAE

Answer:

Find the value of the variable. Justify your answer.

Question 9.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 286

Answer:
x = 5

Explanation:
5x – 4 = 3x + 6
5x – 3x = 6 + 4
2x = 10
x = 5

Question 10.
Big Ideas Math Geometry Answer Key Chapter 10 Circles 287

Answer:
r = 9

Explanation:
(6 + r)² = 12² + r²
36 + 12r + r² = 144 + r²
12r = 108
r = 9

Question 11.
Prove or disprove that the point (2√2, – 1) lies on the circle centered at (0, 2) and containing the point (- 1, 4).

Answer:
Disproved

Explanation:
We consider the circle centred at the A(0, 2) and containing the point B(-1, 4).
AB = √(-1 – 0)² + (4 – 2)² = √1 + 4 = √5
The distance between centre A(0, 2) and P(2√2, – 1) is
AP = √(2√2 – 0)² + (-1 – 2)² = √8 + 9 = √17
AB ≠ AP
So, the point (2√2, – 1) dies not lie on the circle.

Prove the given statement.

Question 12.
\(\widehat{S T} \cong \widehat{R Q}\)
Big Ideas Math Geometry Answer Key Chapter 10 Circles 288

Answer:

Question 13.
\(\widehat{J M} \cong \widehat{L M}\)
Big Ideas Math Geometry Answer Key Chapter 10 Circles 289

Answer:

Question 14.
\(\widehat{D G} \cong \widehat{F G}\)
Big Ideas Math Geometry Answer Key Chapter 10 Circles 290

Answer:

Question 15.
A bank of lighting hangs over a stage. Each light illuminates a circular region on the stage. A coordinate plane is used to arrange the lights, using a corner of the stage as the origin. The equation (x – 13)2 + (y – 4)2 = 16 represents the boundary of the region illuminated by one of the lights. Three actors stand at the points A(11, 4), B(8, 5), and C(15, 5). Graph the given equation. Then determine which actors are illuminated by the light.

Answer:
The equation (x – 13)² + (y – 4)²= 16 represents the standard equation of the circle with center (13, 4) and radius 4
Graph the circle with center S(13, 4), radius 4. Then graph the points A(11,4), B (8, 5), C(15,5) which represents the places where the actors stand.
Big Ideas Math Geometry Answers Chapter 10 Circles 24
From the graph, we can see that points A and C inside the circle and point B is outside the circle. Therefore, actors who stand at points A and C are illuminated by the light

Question 16.
If a car goes around a turn too quickly, it can leave tracks that form an arc of a circle. By finding the radius of the circle, accident investigators can estimate the speed of the car.
Answer:

Big Ideas Math Geometry Answer Key Chapter 10 Circles 291

a. To find the radius, accident investigators choose points A and B on the tire marks. Then the investigators find the midpoint C of \(\overline{A B}\). Use the diagram to find the radius r of the circle. Explain why this method works.

Answer:
The radius r of the circle = 155.71 ft

Explanation:
Given that,
AC = 130 ft, CD = 70 ft
CE = (r – 70) ft
r² = a² + b²
r²= 130²+ (r – 70)²
r² = 16900 + r² – 140r + 4900
140r = 21,800
r = 155.71 ft

b. The formula S = 3.87√fr can be used to estimate a car’s speed in miles per hour, where f is the coefficient of friction and r is the radius of the circle in feet. If f = 0.7, estimate the car’s speed in part (a).

Answer:
The estimated car’s speed is 39.67 miles per hour

Explanation:
S = 3.87√fr
S = 3.8 √(0.7 x 155.71)
S = 3.8 √108.997
S = 3.8 x 10.44
S = 39.67

Circles Cumulative Assessment

Question 1.
Classify each segment as specifically as possible.
Big Ideas Math Geometry Solutions Chapter 10 Circles 292
a. \(\overline{B G}\)

Answer:
\(\overline{B G}\) is a chord

b. \(\overline{C D}\)

Answer:
\(\overline{C D}\) is radius.

c. \(\overline{A D}\)

Answer:
\(\overline{A D}\) is diameter.

d. \(\overline{F E}\)

Answer:
\(\overline{F E}\) is a chord

Question 2.
Copy and complete the paragraph proof.

Big Ideas Math Geometry Solutions Chapter 10 Circles 293

Given Circle C with center (2, 1) and radius 1,
Circle D with center (0, 3) and radius 4
Prove Circle C is similar to Circle D.

Map Circle C to Circle C’ by using the _________ (x, y) → _________ so that Circle C’ and Circle D have the same center at (____, _____). Dilate Circle C’ using a cellIer of dilation (_____, _____) and a scale factor of _____ . Because there is a _________ transformation that maps Circle C to Circle D, Circle C is __________ Circle D.

Answer:
Map Circle C to Circle C’ by using the scale factor (x, y) → (0, 3) so that Circle C’ and Circle D have the same center at (2, 1). Dilate Circle C’ using a cellIer of dilation (2, 1) and a scale factor of circles. Because there is a transformation that maps Circle C to Circle D, Circle C is similar to Circle D.

Question 3.
Use the diagram to write a proof.
Big Ideas Math Geometry Solutions Chapter 10 Circles 294
Given ∆JPL ≅ ∆NPL
\(\overline{P K}\) is an altitude of ∆JPL
\(\overline{P M}\) is an altitude ∆NPL
Prove ∆PKL ~ ∆NMP

Answer:
∆JPL is similar to ∆NPL and PK is the altitude of ∆JPL and PM is an altitude of ∆NPL
Altitude is a line drawn from one vertex to the opposite site. It is perpendicular to the side.
So, ∆PKL is similar to ∆NMP

Question 4.
The equation of a circle is x² + y² + 14x – 16y + 77 = 0. What are the center and radius of the circle?
(A) center: (14, – 16). radius: 8.8
(B) center: (- 7, 8), radius: 6
(C) center (- 14, 16), radius: 8.8
(D) center: (7, – 8), radius: 5.2

Answer:
(B) center: (- 7, 8), radius: 6

Explanation:
x² + y² + 14x – 16y + 77 = 0
x² + 14x + 49 + y² – 16y + 64 = 36
(x + 7)² + (y – 8)² = 6²
So, the centre is (-7, 8) and radius is 6.

Question 5.
The coordinates of the vertices of a quadrilateral are W(- 7, – 6), X(1, – 2), Y(3, – 6) and Z(- 5, – 10). Prove that quadrilateral WXYZ is a rectangle.

Answer:
Proved

Explanation:
Find the distance of WY and ZX
WY = √(-7 – 3)² + (-6 + 6)² = √(-10)² = 10
ZX = √(1 + 5)² + (-2 + 10)² = √6² + 8² = 10
WY = ZX, the diagonals are congruent
Use the slope formula to find the slopes of diagonals
Slope of WY = \(\frac { -6 + 6 }{ -7 – 3 } \) = 0
Slope of ZX = \(\frac { -2 + 10 }{ 1 + 5 } \) = \(\frac { 4 }{ 3 } \)
Because the product of slopes of diagonals is 0, the diagonals are not perpendicular
Therefore, the quadrilateral WXYZ is a rectangle.

Question 6.
Which angles have the same measure as ∠ACB? Select all that apply.
im – 295
∠DEF ∠JGK ∠KGL ∠LGM ∠MGJ
∠QNR ∠STV ∠SWV ∠VWU ∠XYZ
Answer:
∠VWU

Question 7.
Classify each related conditional statement based on the conditional statement
“If you are a soccer player. then you are an athlete.”
a. If you are not a soccer player, then you are not an athlete.

Answer:
False

b. If you are an athlete, then you are a soccer player.

Answer:
False

c. You are a soccer player if and only if you are an athlete.

Answer:
True

d. If you are not an athlete, then you are not a soccer player.

Answer:
False

Question 8.
Your friend claims that the quadrilateral shown can be inscribed in a circle. Is your friend correct? Explain our reasoning.
Big Ideas Math Geometry Solutions Chapter 10 Circles 296

Answer:
If the sum of any two angles is 180°, then the quadrilateral is inscribed in a circle.
So, 70° + 110° = 180°, 110° + 70°= 180°
So, my friend is correct.

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