I know it seems like there’s a million different things you have to memorize for math class and the distance formula is just another one. Here’s the secret about the distance formula though… it’s not some new thing to learn… you already know it! Don’t believe me? Check this out.

Remember the Pythagorean Theorem? Of course you do! Hmmmm. Okay maybe you don’t. Here it is: in any right triangle with legs of length *a*, *b* and hypotenuse *c*, a^{2 }+ b^{2 }= c^{2}. Okay, now if we solve this equation for *c*, we get

Right? Right. Okay, now think about this here triangle for a sec:

The length of the hypotenuse is d, which is also the distance between the points (x1,y1) and (x2,y2). The length of the legs of this triangle are x2-x1 and y2-y1. So let’s use this information and substitute these values into the Pythagorean Theorem. The hypotenuse is d=c, and the legs are a=x2-x1 and b=y2-y1.

Oh my goodness! It’s amazing! As you can see, the distance formula is just the Pythagorean Theorem in disguise! So go ahead and fall in love… all over again.

For more help, check out our study video for the distance formula.