Slide 9-1 Copyright © 2014 Pearson Education, Inc. 5.4 The Pythagorean Theorem CHAPTER 5

Slide 9-2 Theorem Pythagorean Theorem Copyright © 2014 Pearson Education, Inc. Theorem If a triangle is a right triangle, then the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

Slide 9-3 Many Proofs of the Pythagorean Theorem Elisha Scott Loomis was an Ohio mathematics professor who attempted to collect every known proof of the Pythagorean theorem. He published them in a book that contains 371 different proofs of the Pythagorean theorem. Copyright © 2014 Pearson Education, Inc.

Slide 9-4 Some Famous People who proved the PT Albert Einstein at age 12 President James Garfield Euclid Copyright © 2014 Pearson Education, Inc.

Slide 9-5 Many Proofs of the Pythagorean Theorem Area Proof Copyright © 2014 Pearson Education, Inc.

Slide 9-6 Pythagorean Triple Copyright © 2014 Pearson Education, Inc. A Pythagorean triple is a set of positive integers a, b, and c that satisfy the equation a 2 + b 2 = c 2. Notice from this equation that c must be the greatest number. Below are some common Pythagorean triples. 3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25

Slide 9-7 Example Finding the Length of the Hypotenuse Find the length of the hypotenuse of ΔABC. Check to see that the side lengths of ΔABC form a Pythagorean triple. Explain why. Solution Use either given form of the Pythagorean Theorem. (leg 1 ) 2 + (leg 2 ) 2 = (hypotenuse) 2 a 2 + b 2 = c 2 Copyright © 2014 Pearson Education, Inc.

Slide 9-8 Example Finding the Length of the Hypotenuse a 2 + b 2 = c 2 a 2 + b 2 = x 2 Also the Pythagorean Theorem = x 2 Substitute 21 for a and 20 for b = x 2 Square 21 and square = x 2 Add. 29 = x The length of the hypotenuse is 29. Copyright © 2014 Pearson Education, Inc.

Slide 9-9 Example Finding the Length of the Hypotenuse The length of the hypotenuse is 29. The side lengths 20, 21, and 29 form a Pythagorean triple because they are positive integers that satisfy a 2 + b 2 = c 2, or = 29 2 or = 841, a true statement. Copyright © 2014 Pearson Education, Inc.

Slide 9-10 Example Finding the Length of the Hypotenuse Find the value of x. Write the answer in simplest radical form. Solution a 2 + b 2 = c x 2 = x 2 = 400 x 2 = 336 Copyright © 2014 Pearson Education, Inc.

Slide 9-11 Example Calculating Placement of a Wire A 50-foot supporting wire is to be attached to a 75- foot antenna. Because of surrounding buildings, sidewalks, and roadways, the wire must be anchored exactly 20 feet from the base of the antenna. How high from the base of the antenna must the wire be attached? Give an exact answer and a one-decimal-place approximation. Copyright © 2014 Pearson Education, Inc.

Slide 9-12 Example Calculating Placement of a Wire Solution A right triangle is formed use the Pythagorean Theorem. Copyright © 2014 Pearson Education, Inc.

Slide 9-13 Theorem Converse of the Pythagorean Theorem Copyright © 2014 Pearson Education, Inc. Theorem If the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle.

Slide 9-14 Example Identifying a Right Triangle A triangle has side lengths 85, 84, and 13. Is the triangle a right triangle? Explain. Solution To determine whether we have a right triangle, we use the Converse of the Pythagorean Theorem and see whether the lengths form a Pythagorean triple. Copyright © 2014 Pearson Education, Inc = 7225 is a true statement. The triangle is a right triangle.

Slide 9-15 Theorem °-45°-90° Triangle Theorem Copyright © 2014 Pearson Education, Inc. Theorem In a 45°-45°-90° triangle, both legs are congruent and the length of the hypotenuse is times the length of a leg. hypotenuse =

Slide 9-16 Example Finding the Length of the Hypotenuse Find the value of each variable. a. Solution z = 9 units since the triangle is isosceles. Copyright © 2014 Pearson Education, Inc.

Slide 9-17 Example Finding the Length of the Hypotenuse Find the value of each variable. b. Solution units since the triangle is isosceles. Copyright © 2014 Pearson Education, Inc.

Slide 9-18 Example Finding the Length of a Leg Multiple Choice Find the value of x. a. 3 b. c. 6 d. Solution Copyright © 2014 Pearson Education, Inc. The correct answer is b.

Slide 9-19 Example Finding Distance A high school softball diamond is a square. The distance from base to base is 60 ft. To the nearest foot, how far does a catcher throw the ball from home plate to second base? Solution Because the distance from base to base along the square is the same, the distance d is the length of the hypotenuse of a 45°-45°-90° triangle. Copyright © 2014 Pearson Education, Inc.

Slide 9-20 Theorem °-60°-90° Triangle Theorem Copyright © 2014 Pearson Education, Inc. Theorem In a 30°-60°-90° triangle, the length of the hypotenuse is twice the length of the shorter leg. The length of the longer leg is times the length of the shorter leg. hypotenuse = longer leg =

Slide 9-21 Example Finding the Length of the Shorter Leg Find the value of d in simplest radical form. Solution We are given the length of the longer leg and are asked to find the length of the shorter leg, so that tells us the formula to start with. Copyright © 2014 Pearson Education, Inc.

Slide 9-22 Example Finding the Length of the Shorter Leg Copyright © 2014 Pearson Education, Inc.

Slide 9-23 Example Applying the 30°-60°-90° Triangle Theorem An artisan makes pendants in the shape of equilateral triangles. The height of each pendant is 18 mm. What is the length s of each side of a pendant to the nearest tenth of a millimeter? Solution The equilateral triangle can be divided into two 30°-60°-90° triangles. The hypotenuse of each 30°-60°-90° triangle is s. The shorter leg is half of s. Copyright © 2014 Pearson Education, Inc.

Slide 9-24 Example Applying the 30°-60°-90° Triangle Theorem Each side of the pendant is about 20.8 mm long. Copyright © 2014 Pearson Education, Inc.