March 12: Distance Formula

The Distance Formula is a rather ugly looking formula.  Almost terrifying in fact.  See?

 

More specifically, given the two points (x₁, y₁) and (x₂, y₂), the distance between these points is given by the formula above.


This is very similar to what we did yesterday when we found the distance using a graph.  No, really, it is!  Here- I'll prove it to you! Let's start with 2 arbitrary points: A and B.

We can run lines down from  A and B to create a right triangle (again exactly what we did yesterday.)  And we already know that a² + b² = c² thanks to Pythagoras.

 When we label the coordinates of points A and B, we can see the the intersection of the two legs we created has the same x-value as Point A and the same y-value as point B.

 So the length of side a is equal the change in x from point B to the intersection,

the length of side b is equal the change in y from point A to the intersection.

a = x₂ − x₁   and    b = y₂ − y₁

Click here for today's homework