Question

You can use a right triangle with leg lengths a and b and hypotenuse length c to prove the Pythagorean Theorem. Show that a² + b² = c².
Step 1: Make a square with an area of c².
How can you use the triangle to make a square with an area of c²?
Make each side of the square the same length as a leg of the triangle.
Make each side of the square the same length as the hypotenuse of the triangle.
b
C
a

Answer 1

To make a square of area c² you must make each side length c — so use the hypotenuse.

Place four congruent right triangles so their right angles sit at the corners of a larger square of side a + b and their hypotenuses meet in the center. The four hypotenuses form a smaller central square whose side length is the hypotenuse c, so its area is c². (Alternatively, reflect/rotate the triangle so four hypotenuses form the boundary of a square of side c.)

This arrangement lets you equate areas (area of big square = 4·(1/2)ab + c²), which leads to (a + b)² = 2ab + c² and hence a² + b² = c².