Students can use the Spectrum Math Grade 8 Answer Key Chapter 5 Lesson 5.7 Defining Pythagorean Theorem as a quick guide to resolve any of their doubts.
Spectrum Math Grade 8 Chapter 5 Lesson 5.7 Defining Pythagorean Theorem Answers Key
The Pythagorean Theorem states that if a triangle is a right triangle, then a2 + b2 = c2, when a and b represent the legs of the triangle and c represents the hypotenuse.
The Pythagorean Theorem:
If a triangle is a right triangle, then a2 + b2 = c2.
Converse of Pythagorean Theorem:
If a2 + b2 = c2, then the triangle is a right triangle.
Complete the table below to prove if each set of sides creates a right triangle.
Question 1-12.
Answer:
The Pythagorean Theorem:
If a triangle is a right triangle, then a2 + b2 = c2.
Converse of Pythagorean Theorem:
If a2 + b2 = c2, then the triangle is a right triangle.
1) a = 3, b = 4 , c =5
a2 + b2 = c2
a2 + b2 = 32 + 42 = 9 + 16 = 25
c2 = 52 = 25
Therefore, a2 + b2 = c2 is true and it makes a right triangle.
2) a = 3, b = 4 , c =6
a2 + b2 = c2
a2 + b2 = 32 + 42 = 9 + 16 = 25
c2 = 62 = 36
Therefore, a2 + b2 = c2 is false and it does not make a right triangle.
3) a = 4, b = 6, c =9
a2 + b2 = c2
a2 + b2 = 42 + 62 = 16 + 36 = 52
c2 = 92 = 81
Therefore, a2 + b2 = c2 is false and it does not make a right triangle.
4) a = 5, b = 12 , c =13
a2 + b2 = c2
a2 + b2 = 52 + 122 = 25 + 144 = 169
c2 = 132 = 169
Therefore, a2 + b2 = c2 is true and it makes a right triangle.
5) a = 6, b = 8 , c =13
a2 + b2 = c2
a2 + b2 = 62 + 82 = 36 + 64 = 100
c2 = 132 = 169
Therefore, a2 + b2 = c2 is false and it does not make a right triangle.
6) a = 7, b = 24 , c = 25
a2 + b2 = c2
a2 + b2 = 72 + 242 = 49 + 576 = 625
c2 = 252 = 625
Therefore, a2 + b2 = c2 is true and it makes a right triangle.
7) a = 7, b = 13, c = 15
a2 + b2 = c2
a2 + b2 = 72 + 132 = 49 + 169 = 218
c2 = 152 = 225
Therefore, a2 + b2 = c2 is false and it does not make a right triangle.
8) a = 8, b = 20 , c = 25
a2 + b2 = c2
a2 + b2 = 82 + 202 = 64 + 400 = 464
c2 = 252 = 625
Therefore, a2 + b2 = c2 is false and it does not make a right triangle.
9) a = 8, b = 15 , c = 17
a2 + b2 = c2
a2 + b2 = 82 + 152 = 64 + 225 = 289
c2 = 172 = 289
Therefore, a2 + b2 = c2 is true and it makes a right triangle.
10) a = 10, b = 27, c = 30
a2 + b2 = c2
a2 + b2 = 102 + 272 = 100 + 729= 829
c2 = 302 = 900
Therefore, a2 + b2 = c2 is false and it does not make a right triangle.
11) a = 13, b = 20 , c = 30
a2 + b2 = c2
a2 + b2 = 132 + 202 = 169 + 400 = 569
c2 = 302 = 900
Therefore, a2 + b2 = c2 is false and it does not make a right triangle.
12) a = 13, b = 21 , c = 29
a2 + b2 = c2
a2 + b2 = 132 + 212 = 169 + 441 = 610
c2 = 292 = 841
Therefore, a2 + b2 = c2 is false and it does not make a right triangle.
Question 13.
Based on the true results in the table above, what pattern can be inferred about the Pythagorean Theorem?
Answer: Based on the true results in the table above, the pattern that can be inferred about the Pythagorean theorem is
a2 + b2 = c2 where a and b represent the legs of the triangle and c represents the hypotenuse.
The Pythagorean Theorem:
If a triangle is a right triangle, then a2 + b2 = c2.
Converse of Pythagorean Theorem:
If a2 + b2 = c2, then the triangle is a right triangle.