We included **HMH Into Math Grade 8 Answer Key PDF** **Module 11 Lesson 1 Prove the Pythagorean Theorem **to make students experts in learning maths.

## HMH Into Math Grade 8 Module 11 Lesson 1 Answer Key Prove the Pythagorean Theorem

I Can prove the Pythagorean Theorem, use the Pythagorean Theorem to find unknown side lengths of right triangles, and identify a Pythagorean triple.

**Spark Your Learning**

The given squares form a right triangle. Find the side lengths and areas of each. What patterns do you notice about the squares and side lengths?

Answer:

Right triangle is 3 cms, 4cms and 5 cms,

Squares, 1. 3 cms, 3 cms, 3 cms,

area = 9 cms, 2. 4 cms, 4 cms, 4 cms,

area = 16 cms, 3. 5 cms, 5 cms,5 cms,

area = 25 cms, Side lengths in squares increased by 1 cm and so areas increased,

Explanation:

Given squares are of lengths and areas of 1. 3 cms, 3 cms, 3 cms,

area = 3 cms X 3 cms = 9 cms, 2. 4 cms, 4 cms, 4 cms,

area = 4 cms X 4 cms = 16 cms, 3. 5 cms, 5 cms,5 cms,

area = 5 cms X 5 cms, 25 cms and the given squares form a right triangle let a,b and c be the side lengths as we know c^{2} = a^{2} + b^{2}, So we have sides as 3 cms and 4 cms we get c^{2} = 3^{2} + 4^{2} , c^{2} = 25,

now value of c is square root of 25 = 5 cms. Side lengths in sqaures increased by 1 cm

and so areas increased.

**Turn and Talk** What types of objects in the real-world are in the shape of a right triangle? Can you find some right triangles in the classroom?

Answer A right angle is a 90 degree angle, so there are so many examples in real life like

Bermuda Triangle, Traffic Signs, Pyramids, Truss Bridges, Sailing Boat, Roof, Staircase and ladder,

Buildings, Monuments, and Towers,The height of a pole or a mountain, Sandwiches or Pizza Slices etc, Yes, some right triangles in the classroom are in the corners of a room, book, cube, windows and

at several other places,

Explanation:

A right angle is an abstraction like most objects in mathematics,

Right angles can be found in some of the most common shapes that we see every day,

Squares, rectangles, and right triangles all have right triangles,

Objects in the real-world are in the shape of a right triangle are Bermuda Triangle, Traffic Signs, Pyramids, Truss Bridges, Sailing Boat, Roof, Staircase and ladder, Buildings, Monuments, and Towers,The height of a pole or a mountain, Sandwiches or Pizza Slices etc, Yes,

some right triangles in the classroom are in the corners of a room, book, cube, windows and

at several other places.

**Build Understanding**

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.

If a and b are the lengths of the legs and c is the length of the hypotenuse, then a^{2} + b^{2} = c^{2}.

Question 1.

Pythagoras was a Greek philosopher and mathematician whose name is given to the Pythagorean Theorem. You can use what you know about similar triangles to prove the Pythagorean Theorem.

A. Using △ABC, draw a line from Point C perpendicular to the hypotenuse. Label the point where this line intersects the hypotenuse as Point D. This breaks the length c into two parts and forms two smaller triangles. Label AD as length e, and label DB as length f. Repeat the labels in the two smaller triangles.

Answer:

Explanation:

Using △ABC, drawn a line from Point C perpendicular to the hypotenuse. Labelled the point where this line intersects the hypotenuse as Point D. This breaks the length c into two parts and forms two smaller triangles.

Labelled AD as length e and label DB as length f. Repeated the labels in the two smaller triangles as shown above.

B. Because of Angle-Angle Similarity, △ABC is similar to △CBD and △ACD. What do you know about the corresponding sides of similar triangles?

Answer:

Two triangles are said to be similar if their corresponding angles are congruent and

the corresponding sides are in proportion,

then the measures of the corresponding angles are equal and the ratios of the lengths of the

corresponding sides are proportional.C. Use similar triangles to compare corresponding hypotenuses and corresponding longer legs in △CBD and △ABC.

Use similar triangles to compare corresponding hypotenuses and corresponding shorter legs in △ABC and △ACD.

Answer:

Explanation:

Used similar triangles to compare corresponding hypotenuses and corresponding longer legs in △CBD and △ABC as a/c = f/a and Used similar triangles to compare corresponding hypotenuses and corresponding shorter legs in △ABC and △ACD as c/b = b/e.

D. Use the Multiplication Property of Equality to rewrite the equations.

cf = __________ and b^{2} = ____________

Answer:

cf = a^{2 }and b^{2} = ce,

Explanation:

Using the Multiplication Property of Equality to rewrite the equations as cf = a^{2 }and b^{2} = ce.

E. Use addition to write a^{2} + b^{2} in terms of c, e, and f.

Then simplify to complete the proof.

Answer:

a^{2} + b^{2} = cf + ce, a^{2} + b^{2} = c(f + e), a^{2} + b^{2} = c^{2},

Explanation:

Using addition to write a^{2} + b^{2} in terms of c, e, and f.

Then simplifying to complete the proof as a^{2} + b^{2} = cf + ce,

a^{2} + b^{2} = c(f + e) as f + e = c so a^{2} + b^{2} = c X c, a^{2} + b^{2} = c^{2}.

**Step It Out**

Question 2.

Use the Pythagorean Theorem to find the length of the set of stairs.

A. Use the equation a^{2} + b^{2} = c^{2}.

Substitute the leg lengths into the equation and simplify.

a^{2} + b^{2} = c^{2}

+ = c^{2}

+ = c^{2}

= c^{2}

Answer:

5^{2} + 12^{2} = c^{2}, 25 + 144 = c^{2}, 169 = c^{2},

Explanation:

Given sides as a = 5 feet, 12 feet, Using the Pythagorean Theorem to find the length of the set of stairs and Using the equation a^{2} + b^{2} = c^{2}.

Substituting the leg lengths into the equation and simplifying as 5^{2} + 12^{2} = c^{2},

25 + 144 = c^{2}, 169 = c^{2}.

B. Take the square root of both sides to solve for c, the hypotenuse.

= c

The length of the set of stairs is _________ feet.

Answer:

Square root 169 = c^{2}, c = 13, The length of the set of stairs is 13 feet,

Explanation:

Taking the square root of both sides to solve for c, the hypotenuse,

Square root 169 = c^{2}, c = 13, The length of the set of stairs is 13 feet.

Question 3.

Only certain right triangles have side lengths that are all integers. The set of integers 5, 12, and 13 is one example of a Pythagorean triple, a set of integers that can form a right triangle. To find another common triple, find the length of the hypotenuse in the triangle shown.

Connect to Vocabulary

A Pythagorean triple is a set of positive integers a, b and c that fits the rule a^{2} + b^{2} = c^{2}.

a^{2} +b^{2} = c^{2}

+ = c^{2}

+ = c^{2}

= c^{2}

= c

The set of integers 3, 4, and ____________ is a common Pythagorean triple.

Answer:

The set of integers 3, 4, and 5 is a common Pythagorean triple,

Explanation:

Given only certain right triangles have side lengths that are all integers. The set of integers 5, 12, and 13 is one example of a Pythagorean triple, a set of integers that can form a right triangle. To find another common triple, finding the length of the hypotenuse in the triangle shown as

3^{2} + 4^{2} = c^{2}, 9 + 16 = c^{2}, 25 = c^{2}, c is equal to the sqaure root of 25 is 5,

therefore the set of integers 3, 4, and 5 is a common Pythagorean triple.

**Turn and Talk** Multiply 3, 4, and 5 each by 2. Is the new set of numbers a Pythagorean triple? Does this hold true for any multiple of 3, 4 and 5?

Answer:

Yes, Yes, for all multiple of 3,4 and 5,

Explanation:

Multiplying 3, 4, and 5 each by 2. We get 6,8 and 10 the new set of numbers are not

a Pythagorean triple as 6^{2} + 8^{2} = c^{2}, 36 + 64 = c^{2}, 25 = c^{2},

Yes this hold true for any multiple of 3, 4 and 5, The set of Pythagorean Triples is endless,

We can prove this with the help of the first Pythagorean Triple (3, 4, 5):

Let n be any interger greater than 1, then 3n, 4n and 5n are also a set of Pythagorean Triple.

This is true because: (3n)2 + (4n)2 = (5n)

Examples:

If n it is (3n,4n,5n)

If 2 it is (6,8,10), If 3 it is (9,12,15) and so on,

So we can make infinitely many triples just using the (3,4,5) triple

Question 4.

Use the Pythagorean Theorem to find the unknown leg length.

Use the equation a^{2} +b^{2} = c^{2}. Substitute the given lengths for one leg and the hypotenuse, and then simplify.

a^{2} + b^{2} = c^{2}

+ b^{2} =

+ b^{2} =

b^{2} =

b =

Answer:

b = 15,

Explanation:

Using the equation a^{2} +b^{2} = c^{2}.

Substituting the given lengths for one leg and

the hypotenuse and then simplifying as

a^{2} + b^{2} = c^{2},

8^{2} + b^{2} = 17^{2},

64 + b^{2} = 289,

b^{2} = 289 – 64 = 225, So b is equal to square root of 225,

we get b = 15.

**Turn and Talk** One leg of a right triangle is 9 inches and the hypotenuse is 21 inches. Find the length of the second leg. Do the lengths of the sides form a Pythagorean triple? Explain your reasoning.

Answer:

The length of the second leg is 18.9

approximately 19,

Yes, the lengths of the sides form a

Pythagorean triple,

Explanation:

Given one leg of a right triangle is 9 inches and

the hypotenuse is 21 inches. Finding the length of

the second leg let it be b as we know substituting as

a^{2} + b^{2} = c^{2},

9^{2} + b^{2} = 21^{2},

81 + b^{2} = 441,

b^{2} = 441 – 81 = 360,

So b is equal to square root of 360,

we get b = 18.97 approximately 19.

Yes the lengths of the sides form a

Pythagorean triple,

As we know Pythagorean triples (a,b,c) are three

non-negative integers that satisfy the condition of

Pythagoras theorem for a right-angled triangle.

As left side 9^{2} + 18.97^{2} = 81 + 359.86 = 440.86 ≈ 441

which is equal to right side therefore the lengths of

the sides form a Pythagorean triple.

**Check Understanding**

Question 1.

Complete the statement about this triangle.

Since the triangle is __________ and the ___________ lengths are 9 and 12 inches, by the Pythagorean Theorem, the hypotenuse length is ________ inches.

Answer:

Since the triangle is right triangle and

the side lengths are 9 and 12 inches, by the Pythagorean

Theorem, the hypotenuse length is 15 inches,

Explanation:

Since the triangle is right triangle and

the side lengths are 9 and 12 inches by the Pythagorean

theorem the hypotenuse length let it be c is

a^{2} + b^{2} = c^{2},

9^{2} + 12^{2} = c^{2},

81 + 144 = c^{2},

c^{2} = 225, So the value of c is square root of 225

which is 15 inches.

Question 2.

A. Find the unknown side length of the right-triangle sail.

Answer:

The unknown side length of the right-triangle sail is 5 feet,

Explanation:

Given the length for one side 12 feet and the hypotenuse 13 feet,

let b be unknown side length we have according to

Pythagorean theorem as a^{2} + b^{2} = c^{2}, So substituting

12^{2} + b^{2} = 13^{2},

144 + b^{2} = 169,

b^{2} = 169 – 144 = 25 , So the value of b is square root of 25

which is 5 feet.

B. Do these lengths form a Pythagorean triple?

Answer:

Yes,

Explanation:

As we know Pythagorean triples (a,b,c) are three

non-negative integers that satisfy the condition of

Pythagoras theorem for a right-angled triangle.

As left side we have 12^{2} + 5^{2} = 144 + 25 = 169

which is equal to right side 13^{2 }=169, therefore the

lengths of the sides form a Pythagorean triple.

**On Your Own**

Question 3.

**Open-Ended** In your own words, describe the Pythagorean Theorem and how you can use it.

Answer:

Pythagorean theorem, the well-known geometric

theorem that the sum of the squares on the legs of

a right triangle is equal to the square on the hypotenuse,

Uses: Architecture and Construction,

Laying out square angles,

Navigation,

Surveying,

Solving two dimensional real life problems,

Explanation:

As we know Pythagorean theorem says the square of

the length of the hypotenuse of a right triangle

equals the sum of the squares of the lengths of the other two sides.

Uses of Pythagorean theorem are in

Architecture and Construction,

Laying out square angles,

Navigation,

Surveying,

Solving two dimensional real life problems,

Examples:

To calculate the length of staircase required to reach a window.

To find the length of the longest item can be kept in your room.

To find the steepness of the hills or mountains.

To find the original height of a tree broken due to

heavy rain and lying on itself.

Question 4.

Find the unknown length.

Answer:

The unknown length is 24 in,

Explanation:

Given the length for one side 7 in and the hypotenuse 25 in,

let b be unknown side length we have according to

Pythagorean theorem as a^{2} + b^{2} = c^{2}, So substituting

7^{2} + b^{2} = 25^{2},

49 + b^{2} = 625,

b^{2} = 625 – 49 = 576 , So the value of b is square root of 576

which is 24 in.

Question 5.

A right triangle has leg lengths of 12 centimeters and 16 centimeters. What is the length of the hypotenuse?

Answer:

The length of the hypotenuse is 20 centimeters,

Explanation:

Given a right triangle side lengthas leg lengths of

12 centimeters and 16 centimeters by the Pythagorean

theorem the hypotenuse length let it be c is

a^{2} + b^{2} = c^{2},

12^{2} + 16^{2} = c^{2},

144 + 256 = c^{2},

c^{2} = 400, So the value of c is square root of 400

which is 20 centimeters.

Question 6.

**Critique Reasoning** Rafael found the unknown side length of the given triangle using 5 the Pythagorean Theorem. Is he correct? Explain.

5^{2} + 13^{2} = c^{2}

25 + 169 = c^{2}

194 = c^{2}

14 ≈ c

Answer:

No, Rafael is incorrect,

The unknown side length is 12 not 14,

Explanation:

Given Rafael found the unknown side length of

the given triangle using the Pythagorean Theorem.

Which is incorrect as instead of finding missing side he

substituted hypotenuse value as missing side the correct

method is let b be the missing side so

5^{2} + b^{2} = 13^{2},

25 + b^{2} =169,

b^{2} = 169 – 25 = 144, b equals to square root of 144 which is 12.

Question 7.

A right triangle has a leg length of 10 meters and a hypotenuse that is 26 meters. What is the length of the other leg?

Answer:

The length of the other leg is 24 meters,

Explanation

Given a right triangle has a leg length of 10 meters and

a hypotenuse that is 26 meters. Let b is the length of

the other leg so 10^{2} + b^{2} = 26^{2},

100 + b^{2} = 676,

b^{2} = 676 – 100 = 576, b equals to square root of 576 which is 24.

Question 8.

**Open-Ended** What is a Pythagorean triple? Give one example.

Answer:

A Pythagorean triple consists of three positive

integers a, b, and c, such that a^{2} + b^{2} = c^{2}.

Such a triple is commonly written (a, b, c) and a

well-known example is (3, 4, 5),

Explanation:

A Pythagorean triple consists of three positive

integers a, b, and c, such that a^{2} + b^{2} = c^{2}.

Such a triple is commonly written (a, b, c) and a

well-known example is (3, 4, 5) as

3^{2} + 4^{2} = 5^{2},

9 + 16 = 25,

25 = 25 as both left side and right side are equal

therefore the set of integers 3, 4, and 5 is

a common Pythagorean triple.

Question 9.

Geography A land surveyor wants to measure the distance across a portion of a lake. The surveyor forms the right triangle shown by forming two legs of a right triangle measured on land. What is the distance across the lake? Show your work.

Answer:

The distance across the lake is 25 miles,

Explanation:

Given geography a land surveyor wants to

measure the distance across a portion of a lake.

The surveyor forms the right triangle shown by

forming two legs of a right triangle measured on land as

20 miles and 15 miles, Applying Pythagorean theorem

a^{2} + b^{2} = c^{2 }, let the distance across the lake be c so

20^{2} + 15^{2} = c^{2},

400 + 225 = c^{2},

625 = c^{2},

So the value of c is square root of 625 which is 25 miles.

Question 10.

A sculpture shaped like a pair of right triangles has leg lengths of 8 feet and 15 feet. What is the length of the hypotenuse of the sculpture?

Answer:

The length of the hypotenuse of the sculpture is 17 feet,

Explanation:

Given a sculpture shaped like a pair of right triangles has

leg lengths of 8 feet and 15 feet. Let c be the length of

the hypotenuse of the sculpture applying Pythagorean theorem

a^{2} + b^{2} = c^{2 },

8^{2} + 15^{2} = c^{2},

64 + 225 = c^{2},

289 = c^{2},

So the value of c is square root of 289 which is 17 feet.

Question 11.

**Critique Reasoning** Ashley claims she found the unknown side length of the triangle. Is she correct? Explain.

18^{2} + 24^{2} = c^{2}

324 + 576 = c^{2}

900 = c^{2}

30 = c

Answer:

No, Ashley is incorrect as the unknown side length is

15.87 ≈ 16 not 30,

Explanation:

Given one side length as 18 and hypotenuse as 24,

Ashley claims she found the unknown side length of

the triangle as 30 which is incorrect as the correct side

length is 18^{2} + b^{2} = 24^{2},

324 + b^{2} = 576,

b^{2} = 576 – 374 = 252, b equals to square root of 252

which is 15.87 approximately 16 not 30. So Ashley is incorrect.

**I’m in a Learning Mindset!**

What constructive feedback would you give Ashley to correct her misconception about the unknown side length of the triangle in Problem 11?

Answer:

Would have substituted the value of hypotenuse value

correctly,

Explanation:

The constructive feedback would I give Ashley to

correct her misconception about the unknown

side length of the triangle in Problem 11 is that

she has substituted the value of hypotenuse in the

place of unknown side length and calculated again new

hypotenuse, if she has founded the unknown side length

substituting the given values correctly in the places,

She would have founded the value of unknown length correctly.

**Lesson 11.1 More Practice/Homework**

Question 1.

**Open-Ended** Draw a right triangle on graph paper. Measure the length of each leg and the hypotenuse. Write an equation using your triangle’s side lengths to show the Pythagorean Theorem holds true for your triangle.

Answer:

Equation:

3^{2} + 4^{2} = 5^{2},

Pythagorean Theorem holds true for my triangle,

Explanation:

Drawn a right triangle on graph paper. Measured

the length of each leg and the hypotenuse.

Wrote an equation using my triangle’s side lengths as

3^{2} + 4^{2} = 5^{2}, Now showing the Pythagorean Theorem

holds true for my triangle as we know the square of

the hypotenuse is equal to the sum of the squares of

the other two sides of a triangle.

As left side we have 3^{2} + 4^{2} = 9 + 16 = 25 which is equal to

right side 5^{2 }= 25. So my equation holds true.

Question 2.

A window shaped like a right triangle has the measurements shown. What is the length of the other leg?

Answer:

The length of the other leg is 12 feet,

Explanation:

Given a window shaped like a right triangle

has the measurements shown with one leg of

9 feet and hypotenuse as 15 feet, let b be the

unknown length of the other leg applying

Pythagorean theorem as 9^{2} + b^{2} = 15^{2},

81 + b^{2} = 225,

b^{2} = 225 – 81 = 144, b equals to square root of 144 which is 12 feet.

Question 3.

A right triangle has leg lengths of 7 inches and 24 inches. What is the length of the hypotenuse?

Answer:

The length of the hypotenuse is 25 inches,

Explanation:

Given a right triangle has leg lengths of 7 inches and 24 inches.

Let c be the length of the hypotenuse applying

Pythagorean theorem as a^{2} + b^{2} = c^{2 },

7^{2} + 24^{2} = c^{2},

49 + 576 = c^{2},

625 = c^{2},

So the value of c is square root of 625 which is 25 inches.

Question 4.

**Math on the Spot** Find each unknown side length.

A.

Answer:

The unknown side length hypotenuse is 10 cm,

Explanation:

Given a right triangle with side lengths as

8 cm and 6 cm, let the unknown hypotenuse side be c is

applying Pythagorean theorem

a^{2} + b^{2} = c^{2 },

8^{2} + 6^{2} = c^{2},

64 + 36 = c^{2},

100 = c^{2},

So the value of c is square root of 100 which is 10 cm.

B.

Answer:

The unknown side length is 24 cm,

Explanation:

Given a right triangle with side lengths as

10 cm and hypotenuse side is 26 cm, Let b be the

unknown side length applying Pythagorean theorem

a^{2} + b^{2} = c^{2 },

10^{2} + b^{2} = 26^{2},

100 + b^{2} = 676,

b^{2}= 676 – 100 = 576,

So the value of b is square root of 576 which is 24 cm.

Question 5.

A. Find the unknown side length of a triangle with leg length 5 yards and hypotenuse length 13 yards.

Answer:

The unknown side length is 12 yards,

Explanation:

Given to find the unknown side length of a triangle

with leg length 5 yards and hypotenuse length 13 yards.

Let unknown side be b applying Pythagorean theorem

a^{2} + b^{2} = c^{2 },

5^{2} + b^{2} = 13^{2},

25 + b^{2} = 169,

b^{2}= 169 – 25 = 144,

So the value of b is square root of 144 which is 12 yards.

B. Find the unknown side length of a triangle with leg length 30 yards and hypotenuse length 78 yards.

Answer:

The unknown side length of a triangle is 72 yards,

Explanation:

Given to find the unknown side length of a triangle

with leg length 30 yards and hypotenuse length 78 yards.

Let the unknown side be b applying Pythagorean theorem

a^{2} + b^{2} = c^{2 },

30^{2} + b^{2} = 78^{2},

900 + b^{2} = 6,084,

b^{2}= 6,084 – 900 = 5,184,

So the value of b is square root of 5,184 which is 72 yards.

C. **Use Structure** How are the lengths of the leg and hypotenuse in Part B related to the lengths of the leg and hypotenuse in Part A?

Answer:

The lengths of the leg and hypotenuse in Part B is

6 multiply of the lengths of the leg and hypotenuse in Part A,

Explanation:

If we see the lengths of the leg and hypotenuse in Part B

related to the lengths of the leg and hypotenuse in Part A as

we divide each by their lengths and hypotenuse we get

part B ( 30, 72, 78) and part A (5, 12, 13) we get

30/5 = 6, 72/12 = 6, 78/13 = 6 therefore the lengths of

the leg and hypotenuse in Part B is 6 multiply of the

lengths of the leg and hypotenuse in Part A.

D. **Use Structure** How is the length of the unknown side in Part B related to the length of the unknown side in Part A?

Answer:

The length of the unknown side in Part B is 6 multiple

of the length of the unknown side in Part A,

Explanation:

We have the length of the unknown side in Part B is 72 yards and

the length of the unknown side in Part A is 12 yards

as 72/12 = 6 or 12 X 6 = 72. So the length of the unknown side in Part B is 6 multiple of the length of the unknown side in Part A.

**Test Prep**

Question 6.

What is the unknown side length?

___________ centimeters

Answer:

The unknown side length is 12 cm,

Explanation:

Given a right triangle with side lengths as 5 cm and hypotenuse side is 13 cm, Let b be the

unknown side length applying Pythagorean theorem a^{2} + b^{2} = c^{2 },

5^{2} + b^{2} = 13^{2}, 25 + b^{2} = 169, b^{2}= 169 – 25 = 144, So the value of b is square root of

144 which is 12 cm.

Question 7.

Which are examples of Pythagorean triples? Select all that apply.

(A) 6, 8, 10

(B) 8, 10, 12

(C) 10, 16, 15

(D) 12, 16, 20

(E) 14, 18, 20

Answer:

(A) 6, 8, 10,

(D) 12, 16, 20,

Explanation:

Given to find the examples of Pythagorean triples as we know a Pythagorean triple consists of three

positive integers a, b, and c, such that a^{2} + b^{2} = c^{2 }, So checking bit (A) 6, 8, 10 we have

6^{2} + 8^{2} = c^{2}, 36 + 64 = c^{2}, 100 = c^{2} and c is sqaure root of 100 which is 10 matches,

Checking with bit (B) 8, 10, 12 we have 8^{2} + 10^{2} = c^{2}, 64 + 100 = c^{2}, 164 = c^{2} and c is sqaure root of 164 which is 12.8 which is not matching with 12, Checking with bit (C) 10, 16, 15 we have

10^{2} + 16^{2} = c^{2}, 100 + 256 = c^{2},

356 = c^{2} and c is sqaure root of 356 which is 18.86 which

is not matching with 15,

Checking with bit (D) 12, 16, 20 we have

12^{2} + 16^{2} = c^{2},

144 + 256 = c^{2},

400 = c^{2} and c is sqaure root of 400 which is 20 matches,

Checking with bit (E) 14, 18, 20 we have

14^{2} + 18^{2} = c^{2},

196 + 324 = c^{2},

520 = c^{2} and c is sqaure root of 520 which is 22.8 which is not matching with 20, therefore the examples of Pythagorean triples are bits (A) 6, 8, 10 and (D) 12, 16, 20.

Question 8.

A right triangle has leg lengths of 12 inches and 9 inches. What is the length of the hypotenuse?

__________ inches

Answer:

15 inches,

Explanation:

Given a right triangle has leg lengths of 12 inches and 9 inches.

Let c be the length of the hypotenuse applying Pythagorean theorem

a^{2} + b^{2} = c^{2}, 12^{2} + 9^{2} = c^{2}, 144 + 81 = c^{2}, 225 = c^{2},

So the value of c is square root of 225 which is 15 inches.

Question 9.

A right triangle has leg lengths of 7 feet and 8 feet. What is the length of the hypotenuse? Round to the nearest tenth.

_________ feet

Answer:

The length of the hypotenuse is approximately 11 feet,

Explanation:

Given a right triangle has leg lengths of 7 feet and 8 feet.

Let c be the length of the hypotenuse applying Pythagorean theorem a^{2} + b^{2} = c^{2 },

7^{2} + 8^{2} = c^{2}, 49 + 64 = c^{2}, 113 = c^{2}, So the value of c is square root of 113 which is 10.63 feet

round to the nearest tenth is 11 feet.

**Spiral Review**

Question 10.

Solve the system of equations by graphing. Check your solution.

y = 1 – x

y = 5 – 2x

Answer:

Point of intersection is at (4,-3), So values of x is 4 and y is -3,

Explanation:

Solving the system of equations y = 1 – x, y = 5 – 2x, y = 1 – x, so y = mx + b means

y = -x + 1, here m is -1 and y – intercept b = 1 and if x = 0, y = 1, x = 1, y = 0,

x = 2, y = -1, if x = 3, y = -2,if x = 4, y = -3, if x = 5,

y = -4, (0,1),(1,0), (2, -1), (3,-2) ,(4,-3),(5,-4),

Now y = 5 – 2x, y = -2x + 5, here m = -2 and y – intercept b = 5 and

if x = 0, y = 1, if x = 1, y= 3, if x = 2, y = 1, if x = 3, y = -1,

if x = 4, y = -3, if x =5, y = -5, So (0,5), (1,3),(2,1),(3,-1),(4,-3),(5,-5),

now pointing on graph above as shown we get point of intersection at (4,-3) now checking in equations as y = 1 – x, substituing x as 4 in y = 1- 4 = -3 which is true, now y as -3 we get -3 = 1 – x, x= 1 + 3 = 4 which is true, now checking in y = 5 – 2x substituting x as 4 in

y = 5 – 2 X 4 = 5 – 8 = -3 which is true and now y as -3 we get -3 = 5 – 2x, 2x = 5 + 3, x = 8/2 = 4 which is true, therefore, values of x is 4 and y is -3.

Question 11.

Order the numbers from least to greatest.

\(\frac{10}{4}\), -3, \(\sqrt {8}\) , 2.7

Answer:

Least to greatest are -3, \(\frac{10}{4}\), 2.7, \(\sqrt {8}\),

Explanation:

Given to find order the numbers from least to greatest as \(\frac{10}{4}\) = 2.5 and \(\sqrt {8}\) = 2.82, So -3 < \(\frac{10}{4}\) < 2.7 < \(\sqrt {8}\),

therefore the order of the numbers from least to greatest are -3, \(\frac{10}{4}\), 2.7, \(\sqrt {8}\).