a + b = c 2 2 2 Many properties apply to triangles. But we must prove it, before we can use it. We can use proofs #9 and #11 from http://www.cut-the-knot.org/pythagoras/index.shtml Proof #9 is fairly simple, = b a a b a b a b c c c c a a a a b b b b Given that the two squares are congruent and all 8 triangles are congruent, prove a + b = c 2 2 2 Since they are equal in area, and the triangles are also equal, we can take 4 from each side... AND THE BALANCE STAYS EQUAL. One of which, the Pythagorean Theorem, states that a + b = c applies to all right triangles. 2 2 2 leg leg As for proof #11, its a bit more challenging. Given that c is the radius of the circle, prove the Pythagorean Theorem using the following diagram. This requires proving another theorem first, the Chord-Chord Product Theorem. The Chord-Chord Product Theorem states that in a circle with two intersecting chords, AE EC = BE ED. A E B C D To prove the Chord-Chord Product Theorem, we'll use similar triangles. First, we'll start with a circle that has two intersecting chords. A B C D E Then, draw 2 auxiliary lines between A, B and C, D. The reason for this is that between 2 points, there is one line. A B C D E Angles AEB and DEC are congruent because they're vertical angles. Angles BAC and CDB are congruent by the Inscribed Angles Theorem which states that if two angles intercept the same arc, they're congruent. Then, by Angle-Angle Similarity, triangles AEB and DEC are similar. Since we have proved the Chord-Chord Product Theorem, we can now use it to prove the Pythagorean Theorem. Because they are similar, we can set up a proportion, AE | ED = | BE EC Then, by the Cross-Products Property,

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).

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# Pythagorean Theorem Proof Project

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