Ideas in Geometry/Instructive examples/Proving the Pythagorean Theorem By Picture

The problem stated that we were allowed to assume triangles 1 & 2 are similar triangles. And on the larger triangle, the two legs (the sides that are not the hypotenuse) are equal. If you reflect the image, you can create a square within a larger square. After reflecting the image, two more similar triangles are formed (making 4 in total) and one more larger triangle is formed, thus creating a tilted square in the middle.

If the outer, smaller triangles are folded in, toward the center of the square, they do not overlap. The only part of the square showing will now be a small square formed by the longer leg (b) of each triangle. Label this length x.

x = b – a

We know this because a, the shorter side of the triangle, and x together are the same length as b, the longer leg of the triangle. If side a is taken away from side b, x is the measurement of the side that remains.

To prove the Pythagorean theorem: a² + b² = c²

c =hypotenuse of the smaller triangles and is also the length of the side of the square

so, c² = the area of the square

Now we are looking for c² , or the area of the square: (Area of the four triangles) + (Area of inner square) = Area _Square = c²

Area _{four triangles} = 4( ½ ab) Area _{inner square} = (b-a)²

Area_square = 4( ½ ab) + (b-a)² = c² = 2ab + b² – 2ab + a² = c² = b² + a² = c²

a² + b² = c²

By looking for the area of the square (c²), we find that a² + b² = c² , which is the Pythagorean Theorem.

Group Members: Jamie Balayti, Christine Yu, Loren Martell, Kara, Katlin, Amy Nadell