Coordinates, Distance, and Midpoint

Midpoint Formula

Midpoint between Two Points

The midpoint between two points with the same xx- or yy-coordinate is the midpoint of a horizontal or vertical line segment connecting those points.
The midpoint of a line segment connecting two points divides the line segment into two equal halves. The midpoint between two points on a number line is the average value of the two points.
To determine the midpoint MM of a line segment on a number line with endpoints having coordinates xx and yy, calculate the average of endpoint xx and endpoint yy.
On the coordinate plane, the same principle is applied when the xx-coordinates or the yy-coordinates of two points are equal.

The Midpoint between Points with an Identical Coordinate

Point AA (5,11)(5, 11) and Point BB (5,4)(5, -4) Point AA (3,3)(-3, 3) and Point BB (5,3)(5, 3)
The yy-coordinate of the midpoint is the average of the yy-values. The xx-coordinate of the midpoint is the average of the xx-values.
(x,y)=(5,11+(4)2)=(5,1142)=(5,72)=(5,3.5)\begin{aligned} (x,y)&=\left(5,\frac{11+(-4)}{2}\right)\\ &=\left(5, \frac{11-4}{2}\right)\\&=\left(5, \frac{7}{2}\right) \\ &=(5, 3.5)\end{aligned}
(x,y)=(3+52,3)=(22,3)=(1,3)\begin{aligned}(x,y)&=\left(\frac{-3+5}{2},3\right)\\ &=\left(\frac{2}{2}, 3\right)\\&=(1,3)\end{aligned}

Applying the Midpoint Formula

The midpoint formula can be used to find the midpoint between any two points in the coordinate plane.

The midpoint formula is used to calculate the coordinates of the midpoint between points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) in the coordinate plane.

The xx-coordinate of point MM is the average of the xx-coordinates of the two points.
x=x1+x22x=\frac{x_1+x_2}{2}
The yy-coordinate of point MM is the average of the yy-coordinates of the two points.
y=y1+y22y=\frac{y_1+y_2}{2}
The result is the midpoint formula:
(x,y)=(x1+x22,y1+y22)(x,y)=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)
The coordinates of midpoint MM can be determined by averaging the xx-coordinates and averaging the yy-coordinates of point AA and point BB.
Step-By-Step Example
Using the Midpoint Formula
Determine the midpoint between point AA (3,3)(-3, 3) and point BB (9,2)(9, -2).
Step 1
Substitute the coordinates of the points in the midpoint formula.
(x,y)=(x1+x22,y1+y22)=(3+92,3+(2)2)\begin{aligned}(x,y)&=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\\ &=\left(\frac{-3+9}{2},\frac{3+(-2)}{2}\right)\end{aligned}
Step 2
Simplify.
(x,y)=(3+92,3+(2)2)=(62,12)=(3,12)\begin{aligned}(x,y)&=\left(\frac{-3+9}{2},\frac{3+(-2)}{2}\right)\\&=\left(\frac{6}{2},\frac{1}{2}\right)\\&=\left(3,\frac{1}{2}\right)\end{aligned}
Solution
The midpoint between point AA (3,3)(-3,3) and point BB (9,2)(9,-2) is (3,12)\left(3,\frac{1}{2}\right).