What is the Pythagorean Theorem?
The Pythagorean theorem is used to calculate the length of a side of a right triangle when the lengths of the other sides are known. It is defined as:
where is the hypothenuse (longest side in the right triangle), and and are the other two sides as shown in the following diagram:
Derivation of the Pythagorean theorem
Let us understand how this formula is derived. If we duplicate and mirror our triangle, we can put them together to get a rectangle where the hypothenuse is the diagonal. Let us duplicate this some more and arrange as shown in the following diagram:
We arranged the rectangles (each consisting of two triangles) such that the diagonals form a square which we highlight in green:
We know that the area of the green square is . Our goal is to find this area in terms of the other sides and . The green square consistes of four triangles and a smaller square. Each triangle, as shown in the following diagram can be calculated as .
We have four triangles in total, which gives us an area of . We still need to find the area of the inner square:
We can see that the length of the side is , so the area of the square is . This means that the area of the larger square which we found to be can also be expressed as the sum of the four triangles and the small square . We can set them equal and simplify:
We derived the Pythagorean theorem. We can use it to calculate the length of a side of the triangle when two other sides are known.
Given a triangle with sides and , find the length of side :