Proving the Pythagorean Theorem

1. Proving the Pythagorean Theorem

2. This sample is an excerpt from the presentation Proving the Pythagorean Theorem in Boardworks High School Geometry, which contains 50 interactive presentations in total.

3. Information

4. The Pythagorean theorem The Pythagorean theorem: In a right triangle, the square of the length of the hypotenuse is equal tothe sum of the square of the lengths of the legs. The area of the largest square is c × c or c2. c2 The areas of the smaller squares are a2 and b2. c a2 a The Pythagorean theorem can be written as: b b2 c2 =a2 + b2

5. Showing the Pythagorean theorem

6. A proof of the Pythagorean theorem

7. 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 All triangles have three altitudes. This figure shows one altitude of a right triangle. Y X The Pythagorean theorem can be proved using altitudes and similar triangles.

8. 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 both right angles by definition: Z by the reflexive property: ∠ZXY ≅ ∠WXY by the AA similarity postulate:  W △XYZ ~ △WXY ∠XYZ and ∠YWZ are both right angles by definition: Y X ∠XZY ≅ ∠WZY by the reflexive property: by the AA similarity postulate:  △XYZ~△WYZ by the transitive property of congruence:  △WXY~△WYZ

9. Proof using similarity (2) Show that a2 + b2 = (c + d)2 using similar triangles. △XYZ~△YWZ~△XWY Z △XYZis similar to △XWY: c d b b2 = d(c+d) = W b c+d a d △XYZis similar to △YWZ: c a a2 = c(c+d) = X a c+d Y b a2 + b2 = c(c+d) + d(c+d) adding the two equations: = c2+ cd + dc + d2 = c2+ 2cd + d2  = (c+d)2 (c+ d) is the hypotenuse of △XYZ, which proves the Pythagorean theorem.