1. Lesson 8-2 The Pythagorean Theorem (page 290) Essential Question How do you use the Pythagorean Theorem?

2. The Pythagorean Theorem One of the best known and most useful theorems in all of mathematics is the Pythagorean Theorem. This theorem was named after Pythagoras, a Greek mathematician and philosopher. Unfortunately, very little is known about Pythagoras because none of his writings have survived. Many of the accomplishments credited to Pythagoras may actually have been accomplishments of his colleagues and successors. Born: 580 BC to 572 BC and Died: 500 BC to 490 BC

3. In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs. Theorem 8-2 Given: ∆ ABC ∠ ACB is a right angle Prove: c 2 = a 2 + b 2 A C B Pythagorean Theorem ab c

4. Geometrically speaking …

5. President James Garfield's Proof of The Pythagorean Theorem Many proofs of this theorem exist, including one by President Garfield. While serving in the House of Representatives, President James Garfield developed his own proof in The Journal of Education (Volume 3 issue161) in 1876. President Garfield studied math at Williams College (in Williamstown, MA) and taught in the public school in Pownal, Vermont. President Garfield may have been joking when he stated about his proof that, "we think it something on which the members of both houses can unite without distinction of the party." Born: November 19, 1831 and Died: September 19,1881

6. Garfield's Proof The 20th president of the United States gave the following proof to the Pythagorean Theorem. He discovered this proof 5 years before he became President. He hit upon this proof in 1876 during a mathematics discussion with some of the members of Congress. It was later published in the New England Journal of Education. The proof depends on calculating the area of a right trapezoid 2 different ways. The first way is by using the area formula of a trapezoid and the second is by summing up the areas of the three right triangles that can be constructed in the trapezoid. He used this trapezoid in developing his proof.

7. In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs. Given: ∆ ABC ∠ ACB is a right angle Prove: c 2 = a 2 + b 2 A C B N Pythagorean Theorem ab c de

8. StatementsReasons 1.∆ ABC; ∠ ACB is a right ∠ Given 2.Draw a ⊥ from C to AB. 3. 4.ce = a 2 ; cd = b 2 5.ce + cd = a 2 + b 2 Addition Property of = 6.c (e + d) = a 2 + b 2 Distributive Property 7.c 2 = a 2 + b 2 Substitution Property Through a point outside a line, there is exactly one line perpendicular to the given line. Given: ∆ ABC ∠ ACB is a right angle Prove: c 2 = a 2 + b 2 A C N ab c d e Alt. drawn to the hyp. of a rt. ∆, each leg is g.m. bet. the hyp. & the seg. of the hyp. that is adj. to that leg. B Means-Extremes Property of Proportions This was #41 from HW on page 289.

9. Here is your chance to PROVE the Pythagorean Theorem! http://www.ies.co.jp/math/java/geo/pytha2/pytha2.html This is the Challenge on page 294. 5 2 1 4 3

10. Example #1 : Find the value of “ x ”.x = _______ 13 12 x 5

11. Example #2 : Find the value of “ x ”.x = _______ 2 x 4

12. Example #3 : Find the value of “ x ”.x = _________ x 6 9 ➤ ➤

13. Example #4 : Find the value of “ x ”.x = _______ 12 13 x 10 13 55

14. Example #5 : Find the value of “ x ”. AC = 12 & BD = 16.x = _______ 10 x 8 A 6 B CD 6 8

15. Example #6 : Find the value of “ x ”.x = __________ x 5 17 125

16. Assignment Written Exercises on pages 292 & 293 DO NOW: 1 to 21 odd numbers, GRADED: 23 to 31 odd numbers, AND 33 to 36 all numbers SHOW YOUR WORK! Prepare for Quiz on Lessons 8-1 & 8-2 How do you use the Pythagorean Theorem?