1. Proving the Distance Formula

2. What is the distance between points A and B?
We can use the Pythagorean Theorem to find the distance.

3. The point A is at (6, 4) and the point B is at (-3, -4).
Drawing lines to create a right triangle gives us:  

5. With the creation of the right triangle, the length of the hypotenuse (AB) is the distance from point A to point B.
The length of AC is the difference in the y-coordinates of the points, or 4 – (-4), which is 8.
The length of BC is the difference in the x-coordinates of the points, or 6 – (-3), or 9.
Use the Pythagorean Theorem to find the length of AB!!

6. Using the Pythagorean Theorem, we get:
a2 + b2 = c = c = c2 145 = c2 so that c ≈ 12.04
So, the length of AB is 12.04, meaning that the distance from A to B is about units.

7. We can now generalize this process for any two points
We can now generalize this process for any two points. We can generalize the points A and B to be at any coordinate, for example A can be at (x1,y1) and B can be (x2,y2).
Since the points are identified in generic terms, we can see the legs of the right angle as y2 – y1 for the vertical length and x2 – x1 for the horizontal length.

8. Now, using the Pythagorean Theorem,  
(x2 – x1)2 + (y2 – y1)2 = (distance)2
solve for distance by taking the square root of both sides…

9. distance = √(x2 – x1)2 + (y2 – y1)2
You have just derived the distance formula for any two points in the coordinate plane!
(Use this formula to plug in Point A, (3,4), and Point B, (-3,-4), and see if your distance is the same as it was when we used the Pythagorean Theorem.)

10. Not only can we use the distance formula to find the distance between Point A and Point B, we can also find the slope of the line using our two points.
This indicates that the direction between these two points follows a line with a slope of 8/9.