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Pythagorean Theorem Proof Project
Transcript of Pythagorean Theorem Proof Project
AE EC = BE ED. The two chords that intersect are NQ and MP. So, by the Chord-Chord Products Theorem, NR RQ = MR RP. By the Substitution Property of Equality, (c+b)(c-b) = a a. We're left with c -cb+cb-b = a after we use FOIL, 2 2 2 Then by simplifying, c -b = a 2 2 2 c = a +b after using the Addition Property of Equality. 2 2 2 This proves the Pythagorean Theorem because it will work regardless of the size of the triangle or circle. * * * * * * * Proofs #9 and #11 are very different. Comparison Proof #9 is geometric, uses congruent triangles, and area while Proof #11 is primarily algebraic. Even if they're very different, both of them prove that... Proof #11 assigns variables to side lengths to set up an equation and uses similar triangles. So we're left with two red squares and one blue square whose areas are as follows. c 2 a b 2 2 = This proves the Pythagorean Theorem because it will work regardless of the size of the triangles or the squares. c = a + b 2 2 2 In #9, congruent triangles are used to express that the remaining areas are equal after the Subtraction Property of Equality is utilized. In #11, the right triangle has intersecting chords as side lengths and utilizes the Chord-Chord Product Theorem (which is proven with similar triangles).