Pre-Calculus, Calculus, Trig, Geometry, Algebra

Distance and Midpoint Formulas

For points (x1, x2) and (y1, y2), the distance between them is this formula.



And to find the midpoint, we use this formula.





Find the distance between A and B.



We will use the formula to find the distance.

We first label which points are x1, x2, y1 and y2.

The coordinates for A is (-4, 1). We will label -4 x2 and 1 y2.

The coordinates for B is (3, -2). We will label 3 x1 and -2 y1.

Now we plug in these values into the formula.





We then reduce the parentheses.



We can now square off the two numbers.



We add within the radical.



Then find the square root.





Find the midpoint of A and B.



We will use the midpoint formula to find the midpoint.

We first label which points are x1, x2, y1 and y2.

The coordinates for A is (-4, 1). We will label -4 x2 and 1 y2.

The coordinates for B is (3, -2). We will label 3 x1 and -2 y1.

Now we plug in these values into the formula.





Add the numerators together and divide each by 2.

Therefore, the midpoint is (-0.5, -0.5)



The midpoint of CD is M(4, 7). Given C(1, 5), find the coordinates of D.

When giving just one end point and a midpoint, we are expected to find the other end point, which in this case is D.

We start by labeling C’s coordinates 1 as x1 and C’s coordinates 5 as y1.

We plug the midpoint coordinates and the values displayed above in this following formula.





You notice this is an ordered pair. With red, they are equal. Just as blue. Let’s start with red.



Multiply both sides by 2.

8 = 1 + x1

Isolate x1 on its same side by subtracting 1 from each side.

7 = x1
or 7 = x

Now we do blue.



Multiply both sides by 2.

14 = 5 + y1

Isolate y1 on its same side by subtracting 5 from each side.

9 = y1
or 9 = y

Therefore, missing coordinates (x, y) = (7, 9)



The midpoint of AB is M(1, 4). Given A(3, -2), find the coordinates of B.

When giving just one end point and a midpoint, we are expected to find the other end point, which in this case is B.

We start by labeling A’s coordinates 3 as x1 and A’s coordinates -2 as y1.

We plug the midpoint coordinates and the values displayed above in this following formula.





You notice this is an ordered pair. With red, they are equal. Just as blue. Let’s start with red.



Multiply both sides by 2.

2 = 3 + x1

Isolate x1 on its same side by subtracting 3 from each side.

-1 = x1
or -1 = x

Now we do blue.



Multiply both sides by 2.

8 = -2 + y1

Isolate y1 on its same side by adding 2 from each side.

10 = y1
or 10 = y

Therefore, missing coordinates (x, y) = (-1, 10)