# The Pythagorean Theorem

A $\,90^\circ$ angle is called a
*right angle*.
A *right triangle* is a triangle
with a $\,90^\circ\,$ angle.

In a right triangle,
the side opposite the $\,90^\circ$ angle is called the
*hypotenuse*
and the remaining two sides
are called the *legs*.

The angles in any triangle add up to $\,180^\circ\,.$

In any triangle, the longest side is opposite the largest angle, and the shortest side is opposite the smallest angle. Thus, in a right triangle, the hypotenuse is always the longest side.

The *Pythagorean Theorem* gives a beautiful
relationship between the lengths of the sides in a
right triangle:
the sum of the squares of the shorter
sides is equal to the square of the hypotenuse.
Furthermore, if a triangle has this kind
of relationship between the lengths of its
sides, then it must be a right triangle!

Have fun with many proofs of the Pythagorean Theorem! This applet (you'll need Java) is one of my favorites. Let it load, then keep pressing ‘Next’.

## Examples

Note: $x\,$ cannot equal $\,-4\,,$ because lengths are always positive.

The $\,3{-}4{-}5\,$ triangle is a well-known right triangle. Multiplying all the sides of a triangle by the same positive number does not change the angles. Thus, if you multiply the sides of a $\,3{-}4{-}5\,$ triangle by any positive real number $\,k\,,$ then you will still have a right triangle. For example, these are all right triangles:

$6{-}8{-}10$ | ($\,k = 2\,$) |

$9{-}12{-}15$ | ($\,k = 3\,$) |

$1.5{-}2{-}2.5$ | ( $\,k = 0.5\,$ ) |

$3\pi{-}4\pi{-}5\pi$ | ($\,k = \pi\,$ ) |

and so on! |