Exercise 68 Page 34

Let's begin by reviewing the Pythagorean Theorem. This theorem tells us that the square of the length of the hypotenuse $c$ of a right triangle equals the sum of the squares of the lengths of the other two sides $a$ and $b.$

We want to explain how it is related to the Distance Formula. Then let's draw our triangle it in a coordinate plane.

In the graph we can see the coordinates of the vertices of the triangle. Notice that we can express the lengths of the legs $a$ and $b$ as a difference in these coordinates. For example, $a$ is the difference in the $y\text{-}$coordinates of points $(x_1,y_1)$ and $(x_1,y_2).$ Also, $b$ is the difference in the $x\text{-}$coordinates of points $(x_1,y_2)$ and $(x_2,y_2).$ Let's substitute these values in the equation $a^2+b^2=c^2.$ Note that we took the positive root of $c^2$ because $c$ is the length of a segment, so it is positive. Now, notice that in the graph $c$ is the distance between points $(x_1,y_1)$ and $(x_2,y_2).$ The resulting equation is a formula for finding this distance.

Therefore, the Pythagorean Theorem and the Distance Formula are the same equation, only they are written in a different way.