Midpoints and Missing Endpoints | Geometry | LevelKite

A segment is the part of a line between two points, called its endpoints. You write the segment with endpoints $A$ and $B$ as $\overline{AB}$ . Two segments are congruent when they have equal length. The midpoint of a segment is the point on the segment that divides it into two congruent segments.

In this lesson, you will find a midpoint from two endpoints, and then work backward to find a missing endpoint. Call a point a midpoint only when a problem states it or the coordinates prove it, because a drawing can only suggest it. Keep coordinates, distances and midpoints exact: leave fractions exact, keep radicals simplified, and label any decimal as an approximation.

Start on a number line. You already know that the distance between the points at $a$ and $b$ is $|a - b|$ . Suppose $a < b$ , and the point at $m$ is the midpoint of the segment from $a$ to $b$ . By the definition of midpoint, $m$ lies between $a$ and $b$ and is the same distance from each, so $|m - a| = |b - m|$ . Because $a < m < b$ , both $m - a$ and $b - m$ are positive, so you can drop the absolute value bars. Then add $m$ to both sides, add $a$ to both sides (Addition Property of Equality), and divide both sides by $2$ (Division Property of Equality).

m - a = b - m \;\Rightarrow\; 2m - a = b \;\Rightarrow\; 2m = a + b \;\Rightarrow\; m = \frac{a + b}{2}

So $m$ is the average of $a$ and $b$ , which means their sum divided by $2$ . The order of the endpoints does not matter, because $a + b = b + a$ . For example, the midpoint of the segment from $-3$ to $9$ is at $\frac{-3 + 9}{2} = \frac{6}{2} = 3$ . Your distance skill confirms it: $3$ is between $-3$ and $9$ , and $|3 - (-3)| = 6$ and $|9 - 3| = 6$ , so the point at $3$ meets the definition of midpoint.

Now move to the coordinate plane. A point's $x$ -coordinate is measured along the horizontal axis, called the $x$ -axis, and its $y$ -coordinate is measured along the vertical axis, called the $y$ -axis. Use the number line result on each axis.

Take Point A at $(-2, 1)$ and Point B at $(6, 5)$ . The average of the $x$ -coordinates is $\frac{-2 + 6}{2} = \frac{4}{2} = 2$ , and the average of the $y$ -coordinates is $\frac{1 + 5}{2} = \frac{6}{2} = 3$ . That gives the point $(2, 3)$ . The figure plots these three stated points and draws no segment. It follows the stated coordinates, not how the picture looks, so the reason that $(2, 3)$ is the midpoint has to come from the numbers.

Point A is at

(-2, 1)

, Point B is at

(6, 5)

, and the point labelled

(2, 3)

has the average of their coordinates.

Here is that reason. From Point A to $(2, 3)$ , the $x$ -coordinate changes by $2 - (-2) = 4$ and the $y$ -coordinate changes by $3 - 1 = 2$ . From $(2, 3)$ to Point B, the changes are again $6 - 2 = 4$ and $5 - 3 = 2$ . The second move repeats the first, so the three points lie on one straight path and the two parts have equal length. By the definition of midpoint, $(2, 3)$ is the midpoint of $\overline{AB}$ , and a segment has exactly one midpoint.

The two moves match for any endpoints, because each change is half of the total change. To see this for endpoints with $x$ -coordinates $x_1$ and $x_2$ , start from their average $\frac{x_1 + x_2}{2}$ . Rewrite $x_1$ as $\frac{2x_1}{2}$ and $x_2$ as $\frac{2x_2}{2}$ , then subtract. The $y$ -coordinates work the same way.

\frac{x_1 + x_2}{2} - \frac{2x_1}{2} = \frac{x_2 - x_1}{2} \quad \text{and} \quad \frac{2x_2}{2} - \frac{x_1 + x_2}{2} = \frac{x_2 - x_1}{2}

Key concept

The Midpoint Formula: average each coordinate

Midpoint Formula: If $M$ is the midpoint of a segment with different endpoints $(x_1, y_1)$ and $(x_2, y_2)$ , then each coordinate of $M$ is the average of the matching endpoint coordinates. This holds for every such segment, including horizontal and vertical ones, and in either endpoint order.

The converse swaps the if and then parts, and it is also true: a point with these coordinates is the midpoint of that segment. You use the formula to find a midpoint or a missing endpoint, and the converse to check a found endpoint. Keep each coordinate exact, with fractions and simplified radicals, and label any decimal as an approximation.

M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)

Worked example

Given: $\overline{CD}$ has endpoints $C(-5, 8)$ and $D(3, -2)$ . Find: the midpoint $M$ of $\overline{CD}$ .

M is the midpoint of segment CD, so use the Midpoint Formula in its stated direction. C and D are different points, so they form a segment. Start by naming the given coordinates.
$x_1 = -5,\quad y_1 = 8,\quad x_2 = 3,\quad y_2 = -2$
Find the x-coordinate of M. The Midpoint Formula makes it the average of the endpoints' x-coordinates, so add them and divide by 2.
$x = \frac{-5 + 3}{2} = \frac{-2}{2} = -1$
Find the y-coordinate of M. The Midpoint Formula makes it the average of the endpoints' y-coordinates, so add them and divide by 2.
$y = \frac{8 + (-2)}{2} = \frac{6}{2} = 3$
Check with the definition of midpoint, using exact values. From C to M, the x-coordinate changes by 4 and the y-coordinate by -5. From M to D, the changes are the same. The second move repeats the first, so M is on segment CD and divides it into two congruent segments.
$-1 - (-5) = 4 = 3 - (-1), \qquad 3 - 8 = -5 = -2 - 3$

Answer

The midpoint of $\overline{CD}$ is $M(-1, 3)$ .

Worked example

Given: $\overline{PQ}$ has endpoints $P(-3, 7)$ and $Q(4, -2)$ . Find: the midpoint $M$ of $\overline{PQ}$ .

Apply the Midpoint Formula to the x-coordinates. The sum is odd, so the average is not a whole number. Leave it as an exact fraction.
$x = \frac{-3 + 4}{2} = \frac{1}{2}$
Apply the Midpoint Formula to the y-coordinates, and again leave the result as an exact fraction.
$y = \frac{7 + (-2)}{2} = \frac{5}{2}$
Check with the definition of midpoint, using exact fractions. The x-changes match and the y-changes match, so the second move repeats the first. That puts M on segment PQ and makes the two parts congruent.
$\frac{1}{2} - (-3) = \frac{7}{2} = 4 - \frac{1}{2}, \qquad \frac{5}{2} - 7 = -\frac{9}{2} = -2 - \frac{5}{2}$

Answer

The midpoint of $\overline{PQ}$ is $M\left(\frac{1}{2}, \frac{5}{2}\right)$ .

Now reverse the question. Suppose you know one endpoint and the midpoint, and you need the other endpoint. The moves you compared earlier tell you what to expect: the move from the known endpoint to the midpoint must repeat from the midpoint to the missing endpoint. In the number line example from earlier, the endpoint $-3$ and the midpoint $3$ are $|3 - (-3)| = 6$ apart, so the other endpoint is $6$ past the midpoint, at $3 + 6 = 9$ .

To get exact coordinates in the plane, use the Midpoint Formula in its stated direction: the problem gives $M$ as the midpoint, so each coordinate of $M$ equals an average. Write the missing coordinates as $x$ and $y$ . Each axis then gives one equation with one unknown, and you solve the two equations separately. This method requires that $M$ is stated to be the midpoint and that the known endpoint is a different point from $M$ . Keep each solution exact.

Worked example

Given: $M(1, 3)$ is the midpoint of $\overline{AB}$ , and endpoint $A$ is at $(-4, 7)$ . Find: the coordinates of endpoint $B$ .

Let B be (x, y). M is given as the midpoint, so by the Midpoint Formula, the average of the x-coordinates of A and B equals the x-coordinate of M.
$\frac{-4 + x}{2} = 1$
Multiply both sides by 2 to clear the fraction (Multiplication Property of Equality).
$-4 + x = 2$
Add 4 to both sides to get x alone (Addition Property of Equality).
$x = 6$
Write the matching equation for the y-coordinates, again by the Midpoint Formula.
$\frac{7 + y}{2} = 3$
Multiply both sides by 2 (Multiplication Property of Equality), then subtract 7 from both sides (Subtraction Property of Equality).
$7 + y = 6 \;\Rightarrow\; y = -1$
Check with the converse of the Midpoint Formula, using exact values. The averages of the coordinates of A and B equal the coordinates of M, so M is the midpoint of segment AB. A and B are also different points, so they form a segment.
$\frac{-4 + 6}{2} = \frac{2}{2} = 1, \qquad \frac{7 + (-1)}{2} = \frac{6}{2} = 3$

Answer

Endpoint $B$ is at $(6, -1)$ .

Point A is at

(-4, 7)

, the given midpoint is labelled

(1, 3)

, and Point B, the endpoint you found, is at

(6, -1)

.

The figure plots the values from this example, with the $x$ -axis horizontal and the $y$ -axis vertical. From Point A to the midpoint, the $x$ -coordinate changes by $1 - (-4) = 5$ and the $y$ -coordinate changes by $3 - 7 = -4$ . From the midpoint to Point B, the changes are again $6 - 1 = 5$ and $-1 - 3 = -4$ . This is the repeated move you expected. The evidence comes from these stated coordinates, not from how the picture looks.

Worked example

Given: $M\left(-\frac{1}{4}, 2\right)$ is the midpoint of $\overline{RS}$ , and endpoint $R$ is at $\left(\frac{3}{2}, -1\right)$ . Find: the coordinates of endpoint $S$ .

Let S be (x, y). M is given as the midpoint, so the Midpoint Formula gives one equation for each coordinate.
$\frac{\frac{3}{2} + x}{2} = -\frac{1}{4} \quad \text{and} \quad \frac{-1 + y}{2} = 2$
Multiply both sides of each equation by 2 (Multiplication Property of Equality).
$\frac{3}{2} + x = -\frac{1}{2} \quad \text{and} \quad -1 + y = 4$
Subtract 3/2 from both sides of the first equation (Subtraction Property of Equality), and add 1 to both sides of the second (Addition Property of Equality).
$x = -\frac{1}{2} - \frac{3}{2} = -\frac{4}{2} = -2 \quad \text{and} \quad y = 4 + 1 = 5$
Check with the converse of the Midpoint Formula, using exact fractions. The averages equal the coordinates of M, so M is the midpoint of segment RS. R and S are also different points, so they form a segment.
$\frac{\frac{3}{2} + (-2)}{2} = \frac{-\frac{1}{2}}{2} = -\frac{1}{4}, \qquad \frac{-1 + 5}{2} = \frac{4}{2} = 2$

Answer

Endpoint $S$ is at $(-2, 5)$ .

Takeaway: For any segment with different endpoints, each coordinate of the midpoint is the average of the matching endpoint coordinates. When you know both endpoints, you average. When you know one endpoint and the midpoint, you set the averages equal to the midpoint's coordinates and solve. Either way, keep every value exact and check your answer from the stated coordinates, not from a picture.

Next, you will find the point that partitions a directed segment in a given ratio, and you will find the perimeter of a polygon in the coordinate plane.