A right triangle has one ninety degree angle, two legs labeled a and b, and the hypotenuse c which is opposite the right angle and is always the longest side. Here are examples in different orientations.
The Pythagorean Theorem states that in a right triangle, the sum of the squares of the lengths of the two legs a and b is equal to the square of the length of the hypotenuse c.
By drawing squares on each side of the right triangle, we see the area on the hypotenuse square equals the sum of the areas of the squares on the two legs. With a three four five triangle, nine plus sixteen equals twenty five.
We can apply the theorem to calculate missing sides. In the first example, with legs six and eight, the hypotenuse is ten. In the second, with hypotenuse thirteen and one leg five, the other leg is twelve.
The theorem helps calculate real world distances. For a rectangle field four by three, the diagonal is five. A ladder one unit from a wall reaching three units high requires a length found by the theorem. The shortest path between two offset points is the hypotenuse.