Find the midpoint of the segment with the following endpoints

Purplemath

Sometimes you need to find the point that is exactly midway between two other points. For instance, you might need to find a line that bisects (divides into two equal halves) a given line segment. This middle point is called the "midpoint". The concept doesn't come up often, but the Formula is quite simple and obvious, so you should easily be able to remember it for later.

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Think about it this way: If you are given two numbers, you can find the number exactly between them by averaging them, by adding them together and dividing by two. For example, the number exactly halfway between 5 and 10 is:

The Midpoint Formula works exactly the same way. If you need to find the point that is exactly halfway between two given points, just average the x-values and the y-values.

• Find the midpoint P between (−1, 2) and (3, −6).

First, I apply the Midpoint Formula; then, I'll simplify:

So the answer is P = (1, −2).

Technically, the Midpoint Formula is the following:

The Midpoint Formula: The midpoint of two points, (x1, y1) and (x2, y2) is the point M found by the following formula:

But as long as you remember that you're averaging the two points' x- and y-values, you'll do fine. It won't matter which point you pick to be the "first" point you plug in. Just make sure that you're adding an x to an x, and a y to a y.

• Find the midpoint P between (6.4, 3) and (−10.7, 4).

I'll apply the Midpoint Formula, and simplify:

So the answer is P = (−2.15, 3.5).

• Find the value of p so that (−2, 2.5) is the midpoint between (p, 2) and (−1, 3).

I'll apply the Midpoint Formula:

The y-coordinates already match. This reduces the problem to needing to compare the x-coordinates, "equating" them (that is, setting them equal, because they must be the same) and solving the resulting equation to figure out what p is. This will give me the value necessary for making the x-values match. So:

So my answer is:

p = −3

URL: https://www.purplemath.com/modules/midpoint.htm

Finding the midpoint of a line segment

If you have the coordinates (x, y) of the endpoints of a line segment, finding the midpoint of the segment is very simple. Just take the average of the two x-coordinates of the endpoints (add them and divide by two) to get the x-coordinate of the midpoint. Then do the same with the y-coordinates.

The midpoint "formula" is given below. It implies that the x coordinate of the midpoint is half way between the x coordinates of the endpoints, and likewise for the y-coordinates. This is proven using triangle congruence below the formula.

Midpoint formula

The coordinates of the midpoint of a line segment on a plane (2-dimensional) are:

$$\text{midpoint} = \left( \frac{x_2 + x_1}{2}, \; \frac{y_2 + y_1}{2} \right)$$

Proof of the midpoint formula

If we take line segment AE (right →) with midpoint C, we can draw all of the dashed lines parallel to the coordinate axes. All vertical lines are || and all horizontal lines are ||. Now because parallel lines yield congruent pairs of corresponding angles (∠ 1 ≅ ∠ 3, ∠ 2 ≅ ∠ 4), we have that ΔABC ≅ ΔCDE by the ASA (angle-side-angle) theorem.

The congruence of those triangles means that segments AB and CD are congruent, thus B bisects AF, and BC ≅ DE, thus D bisects EF.

So the coordinates of C are x = the midpoint of AF (the average of the x-coordinates of A and E) and y = midpoint of ED (the average of the y-coordinates of A and B.

X

ASA

"Angle-Side-Angle" theorem

If two angles and the connecting side of one triangle are congruent to the corresponding two angles and connecting side of another, then the two triangles are identical

Practice problems

Calculate the midpoint of the segments with the following endpoints:

7.  Find the other endpoint of the segment with endpoint   (-3, -4)   and midpoint   (2, 1).

The midpoint is

$$\left( \frac{-3 -x}{2}, \frac{-4 -y}{2} \right) = (2, 1)$$

This gives us an equation for the x-coordinate of the missing point, and one for the y. They are:

$$ \begin{align} \frac{-3 +x}{2} &= 2 \\ \\ -3 +x &= 4 \\ x &= 7 \end{align}$$

$$ \begin{align} \frac{-4 +y}{2} &= 1 \\ \\ -4 +y &= 2 \\ y &= 6 \end{align}$$

The point is (7, 6)

Extension to lines in 3 and higher dimensions

The midpoint idea can easily be extended to lines in three dimensions, where each point has a three-dimesional coordinate, like (x, y, z) or (-1, 2, 7). It looks like this:

$$\text{mp} = \left( \frac{x_1 + x_2}{2}, \; \frac{y_1 + y_2}{2}, \; \frac{z_1 + z_2}{2} \right)$$

And we can extend that to lines in many dimensions. Even though the look of such line segments might be difficult for you to imagine, they do exist, and each has a midpoint given by:

$$\text{mp} = \left( \frac{x_1 + x_2}{2}, \; \frac{y_1 + y_2}{2}, \; \frac{z_1 + z_2}{2}, \; \dots \right)$$

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