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a 2 + b 2 = c 2

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. Example 1A:.

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a 2 + b 2 = c 2

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  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. a2 + b2 = c2

  2. Example 1A: Find the value of x. Give your answer in simplest radical form. a2 + b2 = c2 22 + 62 = x2 40 = x2 Example 1B: Find the value of x. Give your answer in simplest radical form. a2 + b2 = c2 –4x+ 20 = 0 (x – 2)2 + 42 = x2 20 = 4x 5 = x x2 – 4x+ 4 + 16 = x2

  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. a2 + b2 = c2 (3w)2 + w2 = 122 10w2 = 144

  4. A set of three nonzero whole numbers a, b, and c such that a2 + b2 = c2 is called a Pythagorean triple. Example 3A: Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain. a2 + b2 = c2 142 + 482 = c2 2500 = c2 50 = c The side lengths are nonzero whole numbers that satisfy the equation a2 + b2 = c2, so they form a Pythagorean triple.

  5. The side lengths do not form a Pythagorean triple because is not a whole number. Example 3B: Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain. a2 + b2 = c2 42 + b2 = 122 b2 = 128

  6. B c a A C b 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.

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

  8. ? c2 = a2 + b2 ? 162 = 122 + 72 ? 256 = 144 + 49 Example 4a Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. 7, 12, 16 Step 1 Determine if the measures form a triangle. 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. 256 > 193 Since c2 > a2 + b2, the triangle is obtuse.

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

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