1. The Pythagorean Theorem is probably the most famous mathematical relationship. As you learned in Lesson 1-6, it states that in a right triangle, the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse. a 2 + b 2 = c 2

2. Example 1A: Find the value of x. Give your answer in simplest radical form. a 2 + b 2 = c 2 2 2 + 6 2 = x 2 40 = x 2 Example 1B: Find the value of x. Give your answer in simplest radical form. a 2 + b 2 = c 2 (x – 2) 2 + 4 2 = x 2 x 2 – 4x + 4 + 16 = x 2 –4x + 20 = 0 20 = 4x 5 = x

3. Example 2: Randy is building a rectangular picture frame. He wants the ratio of the length to the width to be 3:1 and the diagonal to be 12 centimeters. How wide should the frame be? Round to the nearest tenth of a centimeter. Let l and w be the length and width in centimeters of the picture. Then l :w = 3:1, so l = 3w. a 2 + b 2 = c 2 (3w) 2 + w 2 = 12 2 10w 2 = 144

4. A set of three nonzero whole numbers a, b, and c such that a 2 + b 2 = c 2 is called a Pythagorean triple. Example 3A: Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain. The side lengths are nonzero whole numbers that satisfy the equation a 2 + b 2 = c 2, so they form a Pythagorean triple. a 2 + b 2 = c 2 14 2 + 48 2 = c 2 2500 = c 2 50 = c

5. Example 3B: Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain. a 2 + b 2 = c 2 4 2 + b 2 = 12 2 b 2 = 128 The side lengths do not form a Pythagorean triple because is not a whole number.

6. The converse of the Pythagorean Theorem gives you a way to tell if a triangle is a right triangle when you know the side lengths. You can also use side lengths to classify a triangle as acute or obtuse. A B C c b a

7. To understand why the Pythagorean inequalities are true, consider ∆ABC. By the Triangle Inequality Theorem, the sum of any two side lengths of a triangle is greater than the third side length. Remember!

8. Example 4a Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. Step 1 Determine if the measures form a triangle. 7, 12, 16 7 + 12 > 16, 7 + 16 > 12, 12 + 16 > 7 so, 7, 12, and 16 can be the side lengths of a triangle. Step 2 Classify the triangle. Since c 2 > a 2 + b 2, the triangle is obtuse. 256 > 193 c 2 = a 2 + b 2 ? 16 2 = 12 2 + 7 2 ? 256 = 144 + 49 ?

9. Example 4b Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. Step 1 Determine if the measures form a triangle. 11, 18, 34 Since 11 + 18 = 29 and 29 > 34, these cannot be the sides of a triangle.