1. Warm Up
Estimate the value of each to the nearest tenth.
1. √30
2. √14
3. √55
4. √48
5.4 or 5.5
3.7 or 3.8
7.4 or 7.5
6.9

2. The Pythagorean Theorem
Learn to use the Pythagorean Theorem and its converse to solve problems.
Course 3

3. Vocabulary
Pythagorean Theorem
leg
hypotenuse

4. Pythagoras was born on the Aegean island of Samos
Pythagoras was born on the Aegean island of Samos. He is best known for the Pythagorean Theorem, which relates the side lengths of a right triangle.

6. Directions: Find the leg of the hypotenuse to the nearest tenth.

7. Example 1 a2 + b2 = c2 Pythagorean Theorem Substitute for a and b.
Find the length of each leg first.
a2 + b2 = c2
Pythagorean Theorem
= c2
Substitute for a and b.
= c2
Simplify powers.
225 = c
Solve for c; c = c2.
15 = c

8. Example 2 c 4 5
a2 + b2 = c2
Pythagorean Theorem
= c2
Substitute for a and b.
= c2
Simplify powers.
41 = c2
41 = c
Solve for c; c = c2.
6.4  c

9. Example 3 Pythagorean Theorem a2 + b2 = c2 Substitute for a and b.5 7 c
Pythagorean Theorem
a2 + b2 = c2
Substitute for a and b.
= c2
Simplify powers.
= c2
Solve for c; c = c2.
74 = c
8.6  c

10. Example 4
a2 + b2 = c2
Pythagorean Theorem 25 b
72 + b2 = 252
Substitute for a and c.
49 + b2 = 625
Simplify powers.
– –49
Subtract 49 from each side.
b2 = 576 7
Find the positive square root
b = 24

11. Example 5
a2 + b2 = c2
Pythagorean Theorem 12 b
42 + b2 = 122
Substitute for a and c.
16 + b2 = 144
Simplify powers.
– –16
Subtract 16 from both sides. 4
b2 = 128
Find the positive square root.
b  11.3

12. Directions: Determine whether or not the given numbers are possible measures for the sides of a right triangle.

13. Example 6
a = 3, b=4 and c = 5

14. Example 7

15. Lesson Summary
Use the figure for Problems 1 and 2.
1. Find the height h of the triangle.
8 m
2. Find the length of side c to the nearest meter.
10 m c h
12 m
3. An escalator in a shopping mall is 40 ft long and 32 ft tall. What distance does the escalator carry shoppers?
6 m
9 m
2624 ≈ 51 ft