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**Apply the Pythagorean Theorem**

Warm Up Lesson Presentation Lesson Quiz

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

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Warm-Up 4. Find x. ANSWER 6 cm

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**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.

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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

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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

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Example 2 SOLUTION = +

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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.

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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

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

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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.

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**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.

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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.

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

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

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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.

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**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.

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

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

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

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

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

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