Pythagorean Theorem

Start by drawing four congruent right triangles with legs a and b, and hypotenuse c. These triangles can be arranged to form two squares, with one square inside the other.

Notice that the side lengths of the outer square are equal to (a+b). Additionally, the side lengths of the inner square are equal to c. The area of both squares and the area of the four triangles are as follows.

Area of the Inner Square	Area of the Outer Square	Area of the Four Triangles
c^2	(a+b)^2	4* ab/2 = 2ab

The area of the outer square equals the sum of the area of the inner square and the area of the four triangles. The previous diagram shows that. (a+b)^2=c^2+2ab This equation can be simplified by expanding the square of the binomial on the left-hand side.

First, draw the altitude from the right angle to the hypotenuse. This divides the hypotenuse into two segments.

Next, apply the Geometric Mean Leg Theorem. Doing this relates the lengths of the legs to the length of the hypotenuse. a^2 = x* c b^2 = y* c These two equations can be added. Then, c can be factored out from the right-hand side. a^2 &= x* c [-0.15cm] ^+ b^2 &= y* c [-0.5ex] [-3ex] a^2+b^2 &= x* c+ y* c a^2+b^2 &=( x+ y)c Drawing the altitude resulted in the outcome that x+ y is equal to c. After substituting this into the equation above and simplifying, the Pythagorean Theorem is obtained. a^2+b^2 &= c* c &⇕ a^2+b^2 &= c^2