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 Lesson Menu

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

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

Key Concept

a2 + b2 = c2 Pythagorean Theorem 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. a2 + b2 = c2 Pythagorean Theorem 122 + 162 = c2 Replace a with 12 and b with 16. 144 + 256 = c2 Evaluate 122 and 162. 400 = c2 Add 144 and 256. Example 1

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

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. 18 + 9 = c; c = 27 cm B. 182 + 92 = c2; c = 20.1 cm C. 182 – 92 = c; c = 243 cm D. 182 – 92 = c2; c = 15.6 cm Example 1 CYP

a2 + b2 = c2 Pythagorean Theorem 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. a2 + b2 = c2 Pythagorean Theorem a2 + 282 = 332 Replace b with 28 and c with 33. a2 + 784 = 1,089 Evaluate 282 and 332. Example 2

a2 + 784 – 784 = 1,089 – 784 Subtract 784 from each side. Find a Missing Length a2 + 784 – 784 = 1,089 – 784 Subtract 784 from each side. a2 = 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. 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. A. 12 + b2 = 37; 5 ft B. 12 + b = 37; 25 ft C. 12 + b2 = 372; 36.8 ft D. 122 + b2 = 372; 35 ft Example 2 CYP

Key Concept 2

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. a2 + b2 = c2 Pythagorean Theorem 242 + 72 = 252 a = 24, b = 7, c = 25 576 + 49 = 625 Evaluate 242, 72, and 252. 625 = 625  Simplify. Answer: The triangle is a right triangle. Example 3

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

Write an equation you could use to find the length of the missing side of the right triangle. Then find the missing length. A. 3 + 4 = x; 7 cm B. 32 + 42 = x; 25 cm C. 32 + 42 = x2; 5 cm D. 42 – 32 = x2; 1 cm Five Minute Check 1

Write an equation you could use to find the length of the missing side of the right triangle. Then find the missing length. A. 15 + x = 25; 10 ft B. 152 + x = 252; 400 ft C. 15 + x2 = 25; 3.1 ft D. 152 + x2 = 252; 20 ft Five Minute Check 2

Write an equation you could use to find the length of the missing side of the right triangle. Then find the missing length. A. x + 12 = 13; 1 in. B. x2 + 122 = 132; 5 in. C. x + 122 = 132; 25 in. D. x2 – 122 = 132; 17.7 in. Five Minute Check 3

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

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