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1 of 8 © Boardworks 2014 This sample is an excerpt from the presentation Proving the Pythagorean Theorem in High School Geometry, which contains 50 interactive presentations in total.2014 2Boardworks of 8 © Boardworks The Pythagorean theorem The Pythagorean theorem: In a right triangle, the square of the length of the hypotenuse is equal to the sum of the square of the lengths of the legs. The area of the largest square is c × c or c2. c2 a2 a c b The areas of the smaller squares are a2 and b2. The Pythagorean theorem can be written as: b2 c2 = a2 + b2 3 of 8 © Boardworks 2014 Showing the Pythagorean theorem 4 of 8 © Boardworks 2014 A proof of the Pythagorean theorem 5 of 8 © Boardworks 2014 An altitude of a triangle What is an altitude of a triangle? Z An altitude of a triangle is a perpendicular line segment from a side to the opposite vertex. W Y X All triangles have three altitudes. This figure shows one altitude of a right triangle. The Pythagorean theorem can be proved using altitudes and similar triangles. 6 of 8 © Boardworks 2014 Proof using similarity (1) Show that the three triangles in this figure are similar. The altitude to the hypotenuse of a right triangle creates three right triangles: △XYZ, △WXY and △WYZ. ∠XYZ and ∠XWY are by definition: Z both right angles by the reflexive property: ∠ZXY ≅ ∠WXY by the AA similarity postulate: △XYZ ~ △WXY by definition: ∠XYZ and ∠YWZ are both right angles Y by the reflexive property: ∠XZY ≅ ∠WZY by the AA similarity postulate: △XYZ ~ △WYZ by the transitive property of congruence: △WXY ~ △WYZ 7 of 8 W X © Boardworks 2014 Proof using similarity (2) Show that a2 + b2 = (c + d)2 using similar triangles. △XYZ ~ △YWZ ~ △XWY Z △XYZ is similar to △XWY: d b = b c+d c b2 = d(c+d) W a d △XYZ is similar to △YWZ: c a = a c+d adding the two equations: a2 = c(c+d) Y b a2 + b2 = c(c+d) + d(c+d) = c2 + cd + dc + d2 = c2 + 2cd + d2 = (c+d)2 (c + d) is the hypotenuse of △XYZ, which proves the Pythagorean theorem. 8 of 8 © Boardworks 2014 X