A right triangle is a triangle with a right angle. In a right triangle, the side opposite the right angle is called the hypotenuse, and the two other sides that make the right angle are called its legs.

Here are some right triangles with the hypotenuse and legs labeled:

If the triangle is a right triangle, then $a$ and $b$ are used to represent the lengths of the legs, and $c$ is used to represent the length of the hypotenuse. The hypotenuse is always the longest side of a right triangle. 

Here are some other right triangles:

Three right triangles are indicated. A square is drawn using each side of the triangles. The triangle on the left has the square labels “a squared equals 16” and “b squared equals 9” attached to each of the legs. The square labeled “c squared equals 25” is attached to the hypotenuse. The triangle in the middle has the square labels “a squared equals 16” and “b squared equals 1” attached to each of the legs. The square labeled “c squared equals 17” is attached to the hypotenuse. The triangle on the right has the square labels “a squared equals 9” and “b squared equals 9” attached to each of the legs. The square labeled “c squared equals 18” is attached to the hypotenuse.

Notice that for these examples of right triangles, the square of the hypotenuse is equal to the sum of the squares of the legs. In the first right triangle in the diagram, $16+9=25$, in the second, $16+1=17$, and in the third, $9+9=18$. Expressed another way, we have:

$\displaystyle a^2+b^2=c^2$

This is a property of all right triangles, not just these examples, and is often known as the Pythagorean Theorem. The name comes from a mathematician named Pythagoras who lived in ancient Greece around 2,500 BCE, but this property of right triangles was also discovered independently by mathematicians in other ancient cultures including Babylon, India, and China. In China, a name for the same relationship is the Shang Gao Theorem.

It is important to note that this relationship does not hold for all triangles. Here are some triangles that are not right triangles. Notice that the lengths of their sides do not have the special relationship $a^2+b^2=c^2$. That is, $16+10$ does not equal 18, and $10+2$ does not equal 16.

Two right triangles are indicated. A square is drawn using each side of the triangles. The triangle on the left has the square labels “a squared equals 16” aligned to the bottom horizontal leg and “b squared equals 10” aligned to the left leg. The square labeled “c squared equals 18 is aligned with the hypotenuse. The triangle on the right has the square labels of “a squared equals 10” aligned with the bottom leg and “b squared equals 2” aligned with the left leg. The square labeled “c squared equals 16” is aligned with the hypotenuse.