Lesson 8-2 The Pythagorean Theorem (page 290) Essential Question How can you apply right triangle facts to solve real life problems?

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

Theorem 8-2 Pythagorean Theorem In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs. C Given: ∆ ABC ∠ACB is a right angle Prove: c2 = a2 + b2 b a A c B

Geometrically speaking …

Pythagorean Theorem 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: c2 = a2 + b2 C b a d e A N B c

∠ACB is a right angle b a d e c C Given: ∆ ABC Prove: c2 = a2 + b2 A N Statements Reasons ∆ ABC; ∠ACB is a right ∠ Given Draw a ⊥ from C to AB. ce = a2 ; cd = b2 ce + cd = a2 + b2 Addition Property of = c (e + d) = a2 + b2 Distributive Property c2 = a2 + b2 Substitution Property Through a point outside a line, there is exactly one line perpendicular to the given line. 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. Means-Extremes Property of Proportions This was #41 from HW on page 289.

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

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.

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

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

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

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

Example #4: Find the value of “x”. x = _______ 12 13 13 x 5 5 10

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

Example #6: Find the value of “x”. x = __________ 5 x 12 5 17

How can you apply right triangle facts to solve real life problems? Assignment Written Exercises on pages 292 to 294 RECOMMENDED: 1 to 11 odd numbers, 33, 34, 35 REQUIRED: 13 to 31 odd numbers, 36 Prepare for Quiz on Lessons 8-1 & 8-2 How can you apply right triangle facts to solve real life problems?