1. Apply the Pythagorean Theorem
Warm Up
Lesson Presentation
Lesson Quiz

2. Warm-Up
1. Solve x2 = 100.
ANSWER
10, –10
2. Solve x2 + 9 = 25.
ANSWER
4, –4
3. Simplify 20.
ANSWER
2 5

3. Warm-Up
4. Find x.
ANSWER
6 cm

4. Find the length of the hypotenuse of the right triangle.
Example 1
Find the length of the hypotenuse of the right triangle.
SOLUTION
(hypotenuse)2
= (leg)2 + (leg)2
Pythagorean Theorem
x2 =
Substitute.
x2 =
Multiply.
x2 = 100
Add.
x = 10
Find the positive square root.

5. Guided Practice
Identify the unknown side as a leg or hypotenuse. Then, find the unknown side length of the right triangle. Write your answer in simplest radical form. 1.
ANSWER
Leg; 4

6. Guided Practice
Identify the unknown side as a leg or hypotenuse. Then, find the unknown side length of the right triangle. Write your answer in simplest radical form. 2.
hypotenuse; 2 13
ANSWER

7. Example 2
SOLUTION = +

8. Example 2
162 = 42 + x2
Substitute.
256 = 16 + x2
Multiply.
240 = x2
Subtract 16 from each side.
240 = x
Find positive square root.
≈ x
Approximate with a calculator.
The ladder is resting against the house at about 15.5 feet above the ground.
ANSWER
The correct answer is D.

9. Guided Practice
The top of a ladder rests against a wall, 23 feet above the ground. The base of the ladder is 6 feet away from the wall. What is the length of the ladder? 3.
about 23.8 ft
ANSWER

10. Guided Practice
The Pythagorean Theorem is only true for what type of triangle? 4.
right triangle
ANSWER

11. Example 3
Find the area of the isosceles triangle with side lengths 10 meters, 13 meters, and 13 meters.
SOLUTION
STEP 1
Draw a sketch. By definition, the length of an altitude is the height of a triangle. In an isosceles triangle, the altitude to the base is also a perpendicular bisector. So, the altitude divides the triangle into two right triangles with the dimensions shown.

12. Use the Pythagorean Theorem to find the height of the triangle. STEP 2
Example 3
Use the Pythagorean Theorem to find the height of the triangle.
STEP 2
c2 = a2 + b2
Pythagorean Theorem
132 = 52 + h2
Substitute.
169 = 25 + h2
Multiply.
144 = h2
Subtract 25 from each side.
12 = h
Find the positive square root.

13. Example 3
STEP 3
Find the area. 1 2
(base) (height)
= (10) (12) = 60 m2 1 2
Area =
The area of the triangle is 60 square meters.

14. Guided Practice
Find the area of the triangle. 5.
ANSWER
about ft2

15. Guided Practice
Find the area of the triangle. 6.
ANSWER
240 m2.

16. Example 4
Find the length of the hypotenuse of the right triangle.
SOLUTION
Method 1: Use a Pythagorean triple.
A common Pythagorean triple is 5, 12, 13. Notice that if you multiply the lengths of the legs of the Pythagorean triple by 2, you get the lengths of the legs of this triangle: 5 2 = 10 and = 24. So, the length of the hypotenuse is = 26.

17. Method 2: Use the Pythagorean Theorem.
Example 4
Method 2: Use the Pythagorean Theorem.
x2 =
Pythagorean Theorem
x2 =
Multiply.
x2 = 676
Add.
x = 26
Find the positive square root.

18. Guided Practice
Find the unknown side length of the right triangle using the Pythagorean Theorem. Then use a Pythagorean triple. 7.
ANSWER
15 in.

19. Guided Practice
Find the unknown side length of the right triangle using the Pythagorean Theorem. Then use a Pythagorean triple. 8.
ANSWER
50 cm.

20. Lesson Quiz 1.
Find the length of the hypotenuse of the right triangle.
ANSWER 39

21. Lesson Quiz 2.
Find the area of the isosceles triangle.
ANSWER
1080 cm 2

22. Lesson Quiz 3.
Find the unknown side length x. Write your answer in simplest radical form.
ANSWER 4 13