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Lesson Menu Main Idea and New Vocabulary NGSSS Key Concept:Pythagorean Theorem Example 1:Find a Missing Length Example 2:Find a Missing Length Key Concept:Converse of Pythagorean Theorem Example 3:Identify a Right Triangle Five-Minute Check

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Main Idea/Vocabulary Use the Pythagorean Theorem. legs hypotenuse Pythagorean Theorem converse

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NGSSS MA.8.G.2.4 Validate and apply Pythagorean Theorem to find distances in real world situations or between points in the coordinate plane.

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Key Concept

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Example 1 Find a Missing Length 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. a 2 + b 2 = c 2 Pythagorean Theorem 12 2 + 16 2 = c 2 Replace a with 12 and b with 16. 144 + 256 = c 2 Evaluate 12 2 and 16 2. 400 = c 2 Add 144 and 256.

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Example 1 Find a Missing Length Answer: So, the hypotenuse is 20 inches long. Definition of square root c = 20 or –20Simplify. The equation has two solutions, 20 and –20. However, the length of a side must be positive.

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Example 1 CYP A.18 + 9 = c; c = 27 cm B.18 2 + 9 2 = c 2 ; c = 20.1 cm C.18 2 – 9 2 = c; c = 243 cm D.18 2 – 9 2 = c 2 ; c = 15.6 cm 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.

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Example 2 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. Find a Missing Length a 2 + b 2 = c 2 Pythagorean Theorem a 2 + 28 2 = 33 2 Replace b with 28 and c with 33. a 2 + 784 = 1,089 Evaluate 28 2 and 33 2.

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Example 2 Find a Missing Length a 2 + 784 – 784 = 1,089 – 784 Subtract 784 from each side. a 2 = 305 Simplify. Definition of square root a 17.5 or –17.5 Use a calculator. Answer: The length of side a is about 17.5 centimeters.

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Example 2 CYP A.12 + b 2 = 37; 5 ft B.12 + b = 37; 25 ft C.12 + b 2 = 37 2 ; 36.8 ft D.12 2 + b 2 = 37 2 ; 35 ft 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.

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Key Concept 2

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Example 3 Identify a Right Triangle The measures of three sides of a triangle are 24 inches, 7 inches, and 25 inches. Determine whether the triangle is a right triangle. a 2 + b 2 = c 2 Pythagorean Theorem 24 2 + 7 2 = 25 2 a = 24, b = 7, c = 25 576 + 49= 625 Evaluate 24 2, 7 2, and 25 2. 625= 625 Simplify. Answer: The triangle is a right triangle.

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Example 3 CYP A.yes B.no The measures of three sides of a triangle are 10 centimeters, 12 centimeters, and 14 centimeters. Determine whether the triangle is a right triangle.

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A.3 + 4 = x; 7 cm B.3 2 + 4 2 = x; 25 cm C.3 2 + 4 2 = x 2 ; 5 cm D.4 2 – 3 2 = x 2 ; 1 cm Write an equation you could use to find the length of the missing side of the right triangle. Then find the missing length. Five Minute Check 1

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A.15 + x = 25; 10 ft B.15 2 + x = 25 2 ; 400 ft C.15 + x 2 = 25; 3.1 ft D.15 2 + x 2 = 25 2 ; 20 ft Write an equation you could use to find the length of the missing side of the right triangle. Then find the missing length. Five Minute Check 2

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A.x + 12 = 13; 1 in. B.x 2 + 12 2 = 13 2 ; 5 in. C.x + 12 2 = 13 2 ; 25 in. D.x 2 – 12 2 = 13 2 ; 17.7 in. Write an equation you could use to find the length of the missing side of the right triangle. Then find the missing length. Five Minute Check 3

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A.yes B.no C.cannot be determined Is a triangle with side lengths of 18, 25, and 33 a right triangle? Five Minute Check 4

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A.12 mi B.33 mi C.35 mi D.50 mi A man drives 33 miles east and 12 miles south. What is the shortest distance between the man and his starting point? Five Minute Check 5

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