How is the distance formula related to the pythagorean theorem?

Answer 1

Take any two points in the plane. Let' s call them P1 and P2 and they have coordinates

(x1,y1) and (x2, y2) respectively.

Now if we want to find the distance between them, we use

the distance formula. But what this formula is really doing is using the pythagorean theorem. Here is why.

You want to find the distance from P1 to P2. Construct a line from x1 to x2 and y1 to y2.

The straight line between P1 and P2 is the hypotenuse of the right triangle you just created. Now, Pythagoras says, that the hypotenuse squared is equal to the sum of the squares of the side. But we need the the length of those sides. The horizontal one is (x2-x1) and the vertical one is (y2-y1). So if we look at (x2-x1)2+(y2-y1)2 this is equal to hypotenuse

of the triangle squared. But the hypotenuse is the distance from P1 to P2.

So if we take the square root of that hypotenuse, we must also take square root

(x2-x1)2+(y2-y1)2 AND this is exactly what the distance formula shows.

It would help to draw a picture to see this.