Objective: To use the Pythagorean Theorem and its converse. Chapter 7 Lesson 2 Objective: To use the Pythagorean Theorem and its converse.

Theorem 7-4   Pythagorean Theorem In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. a2 + b2 = c2

A Pythagorean triple is a set of nonzero whole numbers a, b, and c that satisfy the equation a2 + b2 = c2. Here are some common Pythagorean triples.                                                                                                                                                                                                                                                 3,4,5 5,12,12 8,15,17 7,24,25

Example 1: Pythagorean Triples Find the length of the hypotenuse of ∆ABC. Do the lengths of the sides of ∆ABC form a Pythagorean triple?                                                                                                                                                      The lengths of the sides, 20, 21, and 29, form a Pythagorean triple.

Example 2: Pythagorean Triple A right triangle has a hypotenuse of length 25 and a leg of length 10. Find the length of the other leg. Do the lengths of the sides form a Pythagorean triple? Not a Pythagorean Triple

Example 3: Using Simplest Radical Form Find the value of x. Leave your answer in simplest radical form.     

Example 4: Using Simplest Radical Form The hypotenuse of a right triangle has length 12. One leg has length 6. Find the length of the other leg. Leave the answer in simplest radical form.

Example 5: Find the area of the triangle. 102+h2=122 100+h2=144 h2=44 A=(1/2)bh A=(1/2)(20)(2√11) A=20√11

Example 6: Find the area of the triangle. 72+b2=(√53)2 49+b2=53 b2=4 √53 cm 7 cm A=(1/2)bh A=(1/2)(7)(2) A=7

Theorem 7-5: Converse of the Pythagorean Theorem If the square of the length of one side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle.

Example 7: Is this triangle a right triangle? c2=a2+b2 852=132+842 7225=169+7056 7225=7225 13 84 85

Example 8: A triangle has sides of lengths 16, 48, and 50. Is the triangle a right triangle? c2=a2+b2 502=162+482 2500=256+2304 2500≠2560 Not a right triangle.

If c2>a2+b2, the triangle is obtuse. Theorem 7-6: If the square of the length of the longest side of a triangle is greater than the sum of the squares of the lengths of the other two sides, the triangle is obtuse. If c2>a2+b2, the triangle is obtuse. c a b

If c2<a2+b2, the triangle is acute. Theorem 7-7:If the squares of the length of the longest side of a triangle is less than the sum of the squares of the lengths of the other two sides, the triangle is acute. If c2<a2+b2, the triangle is acute. b a c

Example 9: The lengths of the sides of a triangle are given. Classify each triangle as acute, obtuse, or right. b.) 12,13,15 152=132+122 225=169+144 225<313 ACUTE a.) 6,11,14 142=62+112 196=36+121 196>157 OBTUSE

Assignment: Page 361 – 363 #1-43