The Pythagorean Theorem states that the length of the hypotenuse squared equals the sum of the squares of the two legs. This is written mathematically as:
To verify this statement, first explicitly expressed by Pythagoreans so many years ago, let’s look at an example.
Consider the right triangle below. Does the Pythagorean Theorem hold for this triangle?
Recall that the Pythagorean Theorem states:
Although it is clear that the theorem holds for this specific triangle, we have not yet proved that the theorem will hold for all right triangles. A simple proof, however, will demonstrate that the Pythagorean Theorem is universally valid.
Proof Based on Similar Triangles
Using this set of similar triangles, we can say that: