# Distance and Midpoint Formulas for Points on the Coordinate Plane

## Distance and Midpoint Formulas for Points on the Coordinate Plane

Find the distance between point A and point B, and then find the coordinates of the midpoint of segment AB. First, we’re going to start by finding the distance between points A and B. The distance formula is the square root of x_1 minus x_2, squared, plus, y_1 minus y_2, squared.

That means we need to know what our x_1, x_2, y_1, y_2 is, so let’s look at our points and find those coordinates. A Is at (negative 4, negative 5), B Is at (2, 3), (and it doesn’t matter which one of these points you choose to be your x_1, y_1, x_2, y_2, as long as you’re consistent) so I’m just going to use A as my x_1, y_1, and I’ll use B as my x_2, y_2.

Now we can substitute into our **distance formula**, so it’s the square root of x_1, negative 4, minus x_2, which is 2, squared, plus, y_1, negative 5, minus y_2, 3, squared, and now we simplify following the rules of **PEMDAS**. I have to start inside my parenthesis, negative 4 minus 2, so I can add the inverse, negative 4 plus negative 2, which is negative 6, squared, plus, (we’ll do the same thing with our second set) negative 5 minus 3, (or add the inverse) negative 5 plus negative 3, which is negative 8, squared, and now I’ll simplify by doing the next step in PEMDAS, exponents.

Negative 6 squared is 36, plus, negative 8 squared is 64, and then we add those together, 36 plus 64 is 100, so the square root of 100, and finally we take the square root, the square root of 100 is 10, so the distance between points A and B is 10. The second thing we needed to do was find the coordinates of the midpoint of segment AB.

The midpoint formula is where you add your x’s, x_1 plus x_2, and divide that by 2, and add your y’s, and divide that by 2. Basically, what you’re doing is you’re finding the average of your x-coordinates and the average of your y-coordinates, because you’re adding two numbers together and then dividing by 2, so you’re finding the average.

Again, we needed those coordinates of our points which we already have, and we’ll plug those into our formula. X_1, negative 4, plus x_2, 2, divided by 2, then y_1, negative 5, plus y_2, 3, also divided by 2. Negative 4 plus 2 is negative 2, divided by 2, negative 5 plus 3 is negative 2, divided by 2.

Negative 2 divided by 2, negative 1, and again, negative 2 divided by 2, negative 1, so that means the coordinates of the midpoint of segment AB is (negative 1, negative 1), so that midpoint would be at (negative 1, negative 1). This point is exactly in the middle of your segment AB.