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Chapter 9 Right Triangles and Trigonometry Section 9.2 Pythagorean Theorem PROVE THE PYTHAGOREAN THEOREM USE THE PYTHAGOREAN THEOREM

The Pythagorean Theorem In a Right triangle, the sum of the squares of the legs is equal to the square of the hypotenuse.

PROVING THE PYTHAGOREAN THEOREM THEOREM 9.4 Pythagorean Theorem In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. b a c c 2 = a 2 + b 2

USE THE PYTHAGOREAN THEOREM True True False False True True

Pythagorean Triples If the sides of a Right triangle are integers, then the sides are known as a Pythagorean Triple 7, 24, 25 Do you know any other ?

USING THE PYTHAGOREAN THEOREM A Pythagorean triple is a set of three positive integers a, b, and c that satisfy the equation c 2 = a 2 + b 2. For example, the integers 3, 4, and 5 form a Pythagorean triple because 5 2 = 32 + 4 2.

(hypotenuse)2 = (leg)2 + (leg)2 Finding the Length of a Hypotenuse 12 x 5 Find the length of the hypotenuse of the right triangle. Tell whether the side lengths form a Pythagorean triple. SOLUTION (hypotenuse)2 = (leg)2 + (leg)2 Pythagorean Theorem x 2 = 5 2 + 12 2 Substitute. x 2 = 25 + 144 Multiply. x 2 = 169 Add. x = 13 Find the positive square root. Because the side lengths 5, 12, and 13 are integers, they form a Pythagorean triple.

(hypotenuse)2 = (leg)2 + (leg)2 Finding the Length of a Leg Many right triangles have side lengths that do not form a Pythagorean triple. x 14 7 Find the length of the leg of the right triangle. SOLUTION (hypotenuse)2 = (leg)2 + (leg)2 Pythagorean Theorem 14 2 = 7 2 + x 2 Substitute. 196 = 49 + x 2 Multiply. 147 = x 2 Subtract 49 from each side. 147 = x Find the positive square root. 49 • 3 = x Use product property. 7 3 = x Simplify the radical.

Yes 5, 12, 13 is a Pythagorean triple Finding the Missing Length SOLUTION (hypotenuse)2 = (leg)2 + (leg)2 Pythagorean Theorem 13 2 = 12 2 + x 2 Substitute. 169 = 144 + x 2 Multiply. 25 = x 2 Subtract 144 from each side. 25 = x Find the positive square root. 5= x Simplify the radical. Yes 5, 12, 13 is a Pythagorean triple

Area and the Right Triangle Area equals ½ altitude times base A=½ab b and a are the same as the legs of a Right triangle.

2 poles are supported by 100ft cables 50 ft from the ground How far are the poles apart?

Indirect Measurement Closure Question SUPPORT BEAM These skyscrapers are connected by a skywalk with support beams. You can use the Pythagorean Theorem to find the approximate length of each support beam.

Closure Question x 2 = (23.26)2 + (47.57)2 x = (23.26)2 + (47.57)2 Indirect Measurement Closure Question 23.26 m 47.57 m x support beams Each support beam forms the hypotenuse of a right triangle. The right triangles are congruent, so the support beams are the same length. x 2 = (23.26)2 + (47.57)2 Pythagorean Theorem x = (23.26)2 + (47.57)2 Find the positive square root. x  52.95 Use a calculator to approximate. The length of each support beam is about 52.95 meters.

HW Multi-Step Pythagorean Theorem Handout