Objective The student will be able to:

1. ObjectiveThe student will be able to: • Use the Pythagorean Theorem to find the hypotenuse of a right triangle • Use the converse of the PT to test side lengths of potential right triangles Designed by Skip Tyler, Varina High School

2. Geometry To find the missing side length of a right triangle by using the Pythagorean Theorem, determine whether a triangle is a right triangle by using the converse of the Pythagorean Theorem

3. Geometry legs hypotenuse Pythagorean Theorem converse

4. What is a right triangle? hypotenuse leg right angle It is a triangle which has an angle that is 90 degrees. The two sides that make up the right angle are called legs. The side opposite the right angle is the hypotenuse. leg

5. Geometry Pythagorean Theorem Words In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. Model Symbols

6. Step-by-Step Example Write an equation you could use to find the length of the missing side of the right triangle. Then find the missing length. Round to the nearest tenth if necessary. 1. a2 + b2 = c2 Pythagorean Theorem 1 122 + 92 = c2 Replace a with 12 and b with 9. 2 144 + 81 = c2 Evaluate 122 and 92. 3 225 = c2 Add 81 and 144. ±√225 = c Definition of square root 4 c = 15 or –15 Simplify. The equation has two solutions, 15 and –15. However, the length of a side must be positive. So, the hypotenuse is 15 inches long. 5 Check: a2 + b2 = c2 6 ? 122 + 92 = 152 ? 144 + 81 = 225 Need Another Example? 225 = 225

7. Need Another Example? Write an equation you could use to find the length of the missing side of the right triangle shown. Then find the missing length. Round to the nearest tenth if necessary. 122 + 162 = c2; 20 in. Answer

8. The Pythagorean Theorem In a right triangle, if a and b are the measures of the legs and c is the hypotenuse, then a2 + b2 = c2. Note: The hypotenuse, c, is always the longest side.

9. Find the length of the hypotenuse if1. a = 12 and b = 16. 122 + 162 = c2 144 + 256 = c2 400 = c2 Take the square root of both sides. 20 = c

10. Find the length of the hypotenuse if2. a = 5 and b = 7. 52 + 72 = c2 25 + 49 = c2 74 = c2 Take the square root of both sides. 8.60 = c

11. Find the length of the hypotenuse given a = 6 and b = 12 • 180 • 324 • 13.42 • 18

12. Geometry Converse of Pythagorean Theorem If the sides of a triangle have lengths a, b, and c units such that , then the triangle is a right triangle.

13. Step-by-Step Example The measures of three sides of a triangle are 5 inches, 12 inches, and 13 inches. Determine whether the triangle is a right triangle. 3. a2 + b2 = c2 Pythagorean Theorem 1 ? 52 + 122 = 132 a = 5, b = 12, c = 13 2 ? 25 + 144 = 169 Evaluate 52, 122, and 132. 3 169 = 169 Simplify. 4 The triangle is a right triangle. 5 Need Another Example?

14. Need Another Example? The measures of three sides of a triangle are 24 inches, 7 inches, and 25 inches. Determine whether the triangle is a right triangle. yes; 72 + 242 = 252 Answer

15. 5. The measures of three sides of a triangle are given below. Determine whether each triangle is a right triangle. , 3, and 8 Which side is the biggest? The square root of 73 (= 8.5)! This must be the hypotenuse (c). Plug your information into the Pythagorean Theorem. It doesn’t matter which number is a or b.

16. Sides: , 3, and 832 + 82 = ( ) 2 9 + 64 = 73 73 = 73 Since this is true, the triangle is a right triangle!! If it was not true, it would not be a right triangle.

17. Determine whether the triangle is a right triangle given the sides 6, 9, and • Yes • No • Purple