1. Section 1 – Apply the Pythagorean Theorem
Unit 6 – Right Triangles
Section 1 – Apply the Pythagorean Theorem

2. Suppose these three squares were made of beaten gold, and you were offered either the one large square or the two small squares. Which option would you choose?

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. Example 2
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.
SOLUTION
(hypotenuse)2
= (leg)2 + (leg)2
Pythagorean Theorem
52 = 32 + x2
Substitute.
25= 9 + x2
Multiply.
x2 = 16
Subtract 9 from both sides
x = 4
Find the positive square root.
Leg;4

6. GUIDED PRACTICE Example 3
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.
SOLUTION
(hypotenuse)2
= (leg)2 + (leg)2
Pythagorean Theorem
x2 =
Substitute.
x2 =
Multiply.
x2 = 52
Add.
x = 2 13
Find the positive square root.
hypotenuse; 2 13

7. Example 4
SOLUTION = +

8. EXAMPLE 2 Example 4 SOLUTION 162 = 42 + x2 256 = 16 + x2 240 = x2
Substitute.
256 = 16 + x2
Multiply.
240 = x2
Subtract 16 from each side.
240 = x
Find positive square root.
≈ x
Approximate with a calculator.
ANSWER
The ladder is resting against the house at about 15.5 feet above the ground.
The correct answer is D.

9. GUIDED PRACTICE
Example 5
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?
SOLUTION = +

10. GUIDED PRACTICE Example 5 x2 = (6)2 + (23)2 x2 = 36 + 529 x2 = 565
Substitute.
x2 =
Multiply.
x2 = 565
Add.
x = 23.77
Approximate with a calculator.
about 23.8 ft
ANSWER

11. GUIDED PRACTICE
Example 7
The Pythagorean Theorem is only true for what type of triangle?
right triangle
ANSWER

12. EXAMPLE 8
Find the area of an isosceles triangle
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.

13. Find the area of an isosceles triangle
EXAMPLE 8
Find the area of an isosceles triangle
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.

14. EXAMPLE 8
Find the area of an isosceles triangle
STEP 3
Find the area. 1 2
(base) (height)
= (10) (12) = 60 m2 1 2
Area =
ANSWER
The area of the triangle is 60 square meters.

15. Example 9
Find the area of the triangle.
SOLUTION
To find area of a triangle, first altitude has to be ascertained.
STEP 1
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.

16. GUIDED PRACTICE STEP 2 c2 = a2 + b2 182 = 152 + h2 324 = 225 + h2
Pythagorean Theorem
182 = h2
Substitute.
324 = h2
Multiply.
99 = h2
Subtract 225 from each side.
9.95 = h
Find the positive square root.

17. GUIDED PRACTICE
STEP 3
Find the area. 1 2
(base) (height)
= = ft2 1 2
Area =
ANSWER
The area of the triangle is ft2.

18. Find the area of the triangle.
GUIDED PRACTICE
Example 10
Find the area of the triangle.
SOLUTION
c2 = a2 + b2
Pythagorean Theorem
262 = h2
Substitute.
676 = h2
Multiply.
576 = h2
Subtract 100 from each side.
24 = h
Find the positive square root.

19. GUIDED PRACTICE 1 2 1 2
Area = (base) (height) = (20) (24) = 240 m2
ANSWER
The area of the triangle is 240 square meters.

20. Pythagorean Triples
A Pythagorean triple is a set of three positive integers a, b, and c
that satisfy the equation

21. The Most Common Pythagorean Triples and Their Multiples

22. Example 11
Find the length of a hypotenuse of a right using two methods. 32 24 x