Lesson Objectives
• Learn how to find the midpoint of a line segment
• Learn how to find the unknown coordinate, given the midpoint

## How to Find the Midpoint of a Line Segment

In this lesson, we want to discuss the midpoint formula. First and foremost, let’s introduce the concept of a line segment. A line segment is just a piece of a line. Unlike a line, it has two endpoints and a defined length. Now, the midpoint is just the point that is equidistant (meaning it has the same distance) from the endpoints of our line segment. In other words, the midpoint will cut the line segment in half. Suppose we have a line segment with endpoints: $$(x_1, y_1), (x_2, y_2)$$ We have plotted the point (x,y) as the midpoint of our line segment. To find the x-value, we know that the distance from x1 to x is the same as the distance from x2 to x. $$x_2 - x=x - x_1$$ Solve for x: $$x=\frac{x_1 + x_2}{2}$$ Notice how we are just finding the average of the x-coordinates from our endpoints. We can do the same thing for y: $$y_2 - y=y - y_1$$ Solve for y: $$y=\frac{y_1 + y_2}{2}$$ Again, we are just finding the average of the y-coordinates from our endpoints. We will use a capital M to denote the midpoint, recall that a lowercase m is used for slope. $$M=\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$ Example 1: Find the midpoint of the line segment with the given endpoints. $$(9, 3), (2, -1)$$ Let's assign the first point to be (x1, y1) and the second point to be (x2, y2). $$x_1=9$$ $$y_1=3$$ $$x_2=2$$ $$y_2=-1$$ Plug into the midpoint formula: $$M=\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$ $$M=\left(\frac{9 + 2}{2}, \frac{3 + (- 1)}{2}\right)$$ $$M=\left(\frac{11}{2}, 1\right)$$ Example 2: Find the unknown x-value, given the midpoint of the line segment. $$(12, 5), (x, 9)$$ $$M=\left(\frac{15}{2}, 7\right)$$ Let's assign the first point to be (x1, y1) and the second point to be (x2, y2). $$x_1=12$$ $$y_1=5$$ $$x_2=x$$ $$y_2=9$$ Here, we only need to solve for the unknown x-value. Recall the midpoint formula: $$M=\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$ $$\frac{15}{2}=\frac{12 + x}{2}$$ Multiply both sides by 2: $$\require{cancel}\cancel{2}\cdot \frac{15}{\cancel{2}}=\frac{x + 12}{\cancel{2}}\cdot \cancel{2}$$ $$15=x + 12$$ Subtract 12 away from each side: $$x + 12 - 12=15 - 12$$ $$x=3$$ Our unknown x-value is 3. Our endpoints for the line segment are given as: $$(12, 5), (3, 9)$$ We can check our result by plugging in a 3 for x2 in the midpoint formula. $$x_1=12$$ $$y_1=5$$ $$x_2=3$$ $$y_2=9$$ Check: $$M=\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$ $$M=\left(\frac{12 + 3}{2}, \frac{5 + 9}{2}\right)$$ $$M=\left(\frac{15}{2}, \frac{14}{2}\right)$$ $$M=\left(\frac{15}{2}, 7\right)$$

#### Skills Check:

Example #1

Find the midpoint of the line segment PQ. $$P: (3, 1), Q: (9, -5)$$

A
$$(1, -3)$$
B
$$(6, -2)$$
C
$$(6, 2)$$
D
$$(-1, 3)$$
E
$$(4, 6)$$

Example #2

Find the midpoint of the line segment PQ. $$P: (1, -9), Q: (3, -12)$$

A
$$\left(2, -\frac{21}{2}\right)$$
B
$$\left(5, -\frac{1}{2}\right)$$
C
$$\left(6, -3\right)$$
D
$$\left(-1, \frac{21}{2}\right)$$
E
$$\left(\frac{21}{2}, 2\right)$$

Example #3

Find the midpoint of the line segment PQ. $$P: (-15, 2), Q: (5, 6)$$

A
$$(4, -5)$$
B
$$(-5, 4)$$
C
$$(5, -4)$$
D
$$(3, 2)$$
E
$$(-1, 2)$$