Midpoint Formula Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Midpoint Formula.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A formula for finding the point exactly halfway between two points in the coordinate plane, by averaging their coordinates.

Finding the midpoint is like finding the average position. If two friends live at different addresses on the same street, the midpoint is the house number exactly halfway between them—the average of their two house numbers. In 2D, you just average both coordinates independently.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: The midpoint of two points is found by averaging their $x$ -values and averaging their $y$ -values.

Common stuck point: The procedure for midpoint formula is the easy part; the trap is dividing only one coordinate by 2. Asking "Am I finding the single point centered between two given points (a location, not a length)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I finding the single point centered between two given points (a location, not a length)?

Worked Examples

Example 1

easy

Find the midpoint of the segment joining

(2, 8)

and

(6, 4)

Find the midpoint M of segment AB.

Answer

M = (4, 6)

First step

The midpoint of a segment is the average of the two endpoints' coordinates:

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

. Averaging 'splits the difference' equally.

Full solution

2
Identify the endpoints: $(x_1, y_1) = (2, 8)$ and $(x_2, y_2) = (6, 4)$ . Compute the averages separately for $x$ and $y$ .
3
Substitute: $M = \left(\frac{2+6}{2},\, \frac{8+4}{2}\right) = \left(\frac{8}{2},\, \frac{12}{2}\right) = (4, 6)$ . Verify: the point $(4,6)$ lies exactly halfway between $(2,8)$ and $(6,4)$ — the distance from $(2,8)$ to $(4,6)$ equals the distance from $(4,6)$ to $(6,4)$ (both equal $\sqrt{8}$ ).

The midpoint is simply the average of the

x

-coordinates and the average of the

y

-coordinates. It represents the exact centre of the line segment.

Example 2

medium

The midpoint of segment

PQ

M(3, -1)

. If

P = (-2, 4)

, find the coordinates of

Q

Midpoint M(3,−1) and endpoint P(−2,4) are given. Find Q.

Example 3

easy

Find the midpoint of

(a, b)

and

(c, d)

Example 4

medium

Find the midpoint in 3D of

(0, 4, -2)

and

(6, 8, 10)

Example 5

medium

A median of a triangle goes from vertex

A(1, 0)

to the midpoint of side

BC

, where

B(5, 4)

and

C(3, 10)

. Find that midpoint.

Find M_BC, the midpoint of BC, where the median from A meets side BC.

Example 6

hard

Point

P

divides segment from

A(0, 0)

B(10, 5)

in ratio

2:3

(with

AP:PB = 2:3

). Find

P

Example 7

hard

Given

A(2, 3)

and the midpoint of

AB

(5, 7)

, find

B

, then verify

AM

equals

MB

using the distance formula.

A(2,3) and midpoint M(5,7) are given. Find B.

Example 8

hard

The midpoints of the three sides of a triangle are

(3, 4)

(5, 6)

(4, 2)

. Find the triangle's vertices.

Example 9

challenge

Prove that the midpoint

M

of a segment from

A(x_1, y_1)

B(x_2, y_2)

is equidistant from

A

and

B

, using the distance formula.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Find the midpoint of the segment joining

(-5, 3)

and

(7, -1)

Find the midpoint M of segment AB.

Example 2

easy

A circle has diameter endpoints

A(-6, 2)

and

B(4, 10)

. Find the coordinates of the centre of the circle.

Find the centre of the circle with diameter AB.

Example 3

easy

Find the midpoint of

(0, 4)

and

(8, 0)

Find the midpoint M of segment AB.

Example 4

easy

Find the midpoint of

(-3, 6)

and

(9, -2)

Find the midpoint M of AB.

Example 5

easy

Find the midpoint of

(2, -5)

and

(2, 5)

Find the midpoint M of vertical segment AB.

Example 6

easy

Find the midpoint of

(1.5, 2.5)

and

(4.5, 7.5)

Example 7

medium

The midpoint of segment

AB

(4, 5)

. If

A = (-2, 3)

, find

B

A(−2,3) and midpoint M(4,5) are given. Find B.

Example 8

medium

Find the center of a circle whose diameter has endpoints

(-4, 1)

and

(6, 9)

Find the center of the circle with diameter AB.

Example 9

medium

A triangle has vertices

A(0, 0)

B(6, 0)

C(2, 8)

. Find the midpoints of all three sides.

Find the midpoints of all three sides of triangle ABC.

Example 10

medium

A segment from

(0, 0)

(12, 0)

is divided by its midpoint, then each half is bisected again. List the three division points.

Example 11

medium

Find the midpoint of

(2/3, 1/4)

and

(4/3, 3/4)

Example 12

hard

Three consecutive vertices of a parallelogram are

A(0, 0)

B(5, 1)

C(7, 4)

. Find the fourth vertex

D

Three vertices of a parallelogram are given. Find vertex D.

Example 13

hard

Show that the diagonals of the quadrilateral with vertices

A(1, 1)

B(7, 2)

C(9, 6)

D(3, 5)

bisect each other, and therefore the quadrilateral is a parallelogram.

Both diagonals share midpoint M. Find it and confirm the quadrilateral is a parallelogram.

Example 14

hard

A segment from

A(1, 2)

B(13, 14)

is divided into 4 equal parts. Find the three division points.

Example 15

challenge

In the parallelogram

A(0, 0)

B(b, 0)

C(b+c, d)

D(c, d)

, show using the midpoint formula that the diagonals share a midpoint.

← Back to Midpoint Formula Formula Explained Practice Mode

Related Concepts

Coordinate Plane Addition Division Coordinate Proofs Midsegment Theorem Perpendicularity

Background Knowledge

These ideas may be useful before you work through the harder examples.

coordinate planeadditiondivision