The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of squares of the other two sides. Here we see a right triangle with legs a and b, and hypotenuse c.
The Pythagorean theorem is written as a squared plus b squared equals c squared. In our example, we have a triangle with legs of length 3 and 4, which gives us a hypotenuse of length 5. Let's verify: 3 squared plus 4 squared equals 9 plus 16, which equals 25, and the square root of 25 is 5.
Here we see the visual proof of the Pythagorean theorem. We construct squares on each side of the right triangle. The red square has area 9, the green square has area 16, and the purple square on the hypotenuse has area 25. Notice that 9 plus 16 equals 25, confirming that the area of the square on the hypotenuse equals the sum of the areas of the squares on the other two sides.
There are many sets of three positive integers that satisfy the Pythagorean theorem, called Pythagorean triples. The most famous is the 3-4-5 triangle. Other common triples include 5-12-13, 8-15-17, and 7-24-25. These triples are useful in construction and engineering because they create perfect right angles using simple whole number measurements.