Midpoint Formula, Rule of Midpoint Formula

Published on 23-Oct-2024

Midpoint Calculator

A midpoint is a point on a line that is equidistant from both endpoints of the line or the arc.

To find the midpoint in a line, measure the distance between the two endpoints of a line and divide it by two. The result should be the mid-point you are looking for.

Midpoint Formula

The midpoint $M$ of a line segment with endpoints $A (x_{1}, y_{1})$ and $B (x_{2}, y_{2})$ is given by the formula:

$M = (\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})$

Example: Suppose the distance of a line is 14cm, dividing the length by two gives us 7cm. Therefore the mid-point should be 7cm from the start.

Suppose coordinates of a line are given. (2,9) and (8,17).

How to find the midpoint:

It’s quite simple. We just need to add the value of the x-axis and divide by 2. In the same way, we need to add the value of the y-axis and divide it by two. The mid-point is (2+8/2,9+17/2), (5,13).

Midpoint Word Problems

Problem 1

Find the midpoint of the line segment connecting the points $A (2, 3)$ and $B (6, 7)$ .

Solution: The midpoint $M$ is given by:

$M = (\frac{2 + 6}{2}, \frac{3 + 7}{2}) = (\frac{8}{2}, \frac{10}{2}) = (4, 5)$

Problem 2

The endpoints of a line segment are $A (- 1, 4)$ and $B (3, 2)$ . What is the midpoint?

Solution: The midpoint $M$ is:

$M = (\frac{- 1 + 3}{2}, \frac{4 + 2}{2}) = (\frac{2}{2}, \frac{6}{2}) = (1, 3)$

Problem 3

Find the midpoint between the points $A (0, 0)$ and $B (10, 10)$ .

Solution: The midpoint $M$ is:

$M = (\frac{0 + 10}{2}, \frac{0 + 10}{2}) = (5, 5)$

Problem 4

Given the points $A (- 3, - 2)$ and $B (1, 4)$ , find the midpoint.

Solution: The midpoint $M$ is:

$M = (\frac{- 3 + 1}{2}, \frac{- 2 + 4}{2}) = (\frac{- 2}{2}, \frac{2}{2}) = (- 1, 1)$

Problem 5

The endpoints of a line segment are $A (5, 10)$ and $B (15, 20)$ . What is the midpoint?

Solution: The midpoint $M$ is:

$M = (\frac{5 + 15}{2}, \frac{10 + 20}{2}) = (\frac{20}{2}, \frac{30}{2}) = (10, 15)$

Problem 6

Find the midpoint of the line segment connecting $A (- 4, 0)$ and $B (2, 8)$ .

Solution: The midpoint $M$ is:

$M = (\frac{- 4 + 2}{2}, \frac{0 + 8}{2}) = (\frac{- 2}{2}, \frac{8}{2}) = (- 1, 4)$

Problem 7

Calculate the midpoint between the points $A (1, 1)$ and $B (7, 5)$ .

Solution: The midpoint $M$ is:

$M = (\frac{1 + 7}{2}, \frac{1 + 5}{2}) = (\frac{8}{2}, \frac{6}{2}) = (4, 3)$

Problem 8

Given the points $A (6, 12)$ and $B (14, 4)$ , find the midpoint.

Solution: The midpoint $M$ is:

$M = (\frac{6 + 14}{2}, \frac{12 + 4}{2}) = (\frac{20}{2}, \frac{16}{2}) = (10, 8)$

Problem 9

Find the midpoint of the line segment connecting $A (3, 8)$ and $B (- 5, 2)$ .

Solution: The midpoint $M$ is:

$M = (\frac{3 - 5}{2}, \frac{8 + 2}{2}) = (\frac{- 2}{2}, \frac{10}{2}) = (- 1, 5)$

Problem 10

The endpoints of a line segment are $A (0, 5)$ and $B (4, 9)$ . What is the midpoint?

Solution: The midpoint $M$ is:

$M = (\frac{0 + 4}{2}, \frac{5 + 9}{2}) = (\frac{4}{2}, \frac{14}{2}) = (2, 7)$

Problem 11

The points $A (2, - 3)$ and $B (6, 1)$ are the endpoints of a line segment. What is the midpoint?

Solution: The midpoint $M$ is:

$M = (\frac{2 + 6}{2}, \frac{- 3 + 1}{2}) = (\frac{8}{2}, \frac{- 2}{2}) = (4, - 1)$

Problem 12

Find the midpoint of the segment connecting the points $A (- 2, 5)$ and $B (4, - 1)$ .

Solution: The midpoint $M$ is:

$M = (\frac{- 2 + 4}{2}, \frac{5 - 1}{2}) = (\frac{2}{2}, \frac{4}{2}) = (1, 2)$

Problem 13

Calculate the midpoint of the line segment joining $A (8, 4)$ and $B (10, 12)$ .

Solution: The midpoint $M$ is:

$M = (\frac{8 + 10}{2}, \frac{4 + 12}{2}) = (\frac{18}{2}, \frac{16}{2}) = (9, 8)$

Problem 14

Given the points $A (1, 7)$ and $B (9, 3)$ , find the midpoint of the line segment.

Solution: The midpoint $M$ is:

$M = (\frac{1 + 9}{2}, \frac{7 + 3}{2}) = (\frac{10}{2}, \frac{10}{2}) = (5, 5)$

Problem 15

Find the midpoint between $A (12, - 6)$ and $B (6, 0)$ .

Solution: The midpoint $M$ is:

$M = (\frac{12 + 6}{2}, \frac{- 6 + 0}{2}) = (\frac{18}{2}, \frac{- 6}{2}) = (9, - 3)$

Problem 16

Calculate the midpoint of the segment connecting $A (0, 0)$ and $B (20, 30)$ .

Solution: The midpoint $M$ is:

$M = (\frac{0 + 20}{2}, \frac{0 + 30}{2}) = (10, 15)$

Problem 17

Given points $A (- 5, 5)$ and $B (5, - 5)$ , find the midpoint.

Solution: The midpoint $M$ is:

$M = (\frac{- 5 + 5}{2}, \frac{5 - 5}{2}) = (\frac{0}{2}, \frac{0}{2}) = (0, 0)$

Problem 18

Find the midpoint of the line segment connecting the points $A (3, 3)$ and $B (9, 15)$ .

Solution: The midpoint $M$ is:

$M = (\frac{3 + 9}{2}, \frac{3 + 15}{2}) = (\frac{12}{2}, \frac{18}{2}) = (6, 9)$

Problem 19

The endpoints of a line segment are $A (10, 2)$ and $B (2, 8)$ . What is the midpoint?

Solution: The midpoint $M$ is:

$M = (\frac{10 + 2}{2}, \frac{2 + 8}{2}) = (\frac{12}{2}, \frac{10}{2}) = (6, 5)$

Problem 20

Find the midpoint of the segment joining the points $A (- 6, 4)$ and $B (4, - 2)$ .

Solution: The midpoint $M$ is:

$M = (\frac{- 6 + 4}{2}, \frac{4 - 2}{2}) = (\frac{- 2}{2}, \frac{2}{2}) = (- 1, 1)$

Problem 21

The endpoints of a line segment are $A (3, - 4)$ and $B (7, 2)$ . What is the midpoint of this segment?

Solution: The midpoint $M$ is:

$M = (\frac{3 + 7}{2}, \frac{- 4 + 2}{2}) = (\frac{10}{2}, \frac{- 2}{2}) = (5, - 1)$

Problem 22

Find the midpoint of the segment connecting points $A (- 1, 5)$ and $B (5, 11)$ .

Solution: The midpoint $M$ is:

$M = (\frac{- 1 + 5}{2}, \frac{5 + 11}{2}) = (\frac{4}{2}, \frac{16}{2}) = (2, 8)$

Problem 23

The coordinates of point $A$ are $(2, 3)$ and the coordinates of point $B$ are $(10, 7)$ . Determine the midpoint of the line segment $A B$ .

Solution: The midpoint $M$ is:

$M = (\frac{2 + 10}{2}, \frac{3 + 7}{2}) = (\frac{12}{2}, \frac{10}{2}) = (6, 5)$

Problem 24

Calculate the midpoint of the line segment connecting points $A (1, 2)$ and $B (- 3, - 6)$ .

Solution: The midpoint $M$ is:

$M = (\frac{1 - 3}{2}, \frac{2 - 6}{2}) = (\frac{- 2}{2}, \frac{- 4}{2}) = (- 1, - 2)$

Problem 25

The points $A (0, 0)$ and $B (8, 8)$ are the endpoints of a line segment. Find the midpoint.

Solution: The midpoint $M$ is:

$M = (\frac{0 + 8}{2}, \frac{0 + 8}{2}) = (\frac{8}{2}, \frac{8}{2}) = (4, 4)$

Problem 26

Given points $A (10, 0)$ and $B (14, 12)$ , calculate the midpoint of the segment.

Solution: The midpoint $M$ is:

$M = (\frac{10 + 14}{2}, \frac{0 + 12}{2}) = (\frac{24}{2}, \frac{12}{2}) = (12, 6)$

Problem 27

The points $A (- 7, 5)$ and $B (1, - 3)$ define a line segment. What is the midpoint of this segment?

Solution: The midpoint $M$ is:

$M = (\frac{- 7 + 1}{2}, \frac{5 - 3}{2}) = (\frac{- 6}{2}, \frac{2}{2}) = (- 3, 1)$

Problem 28

Calculate the midpoint of the line segment joining points $A (- 2, - 1)$ and $B (4, 5)$ .

Solution: The midpoint $M$ is:

$M = (\frac{- 2 + 4}{2}, \frac{- 1 + 5}{2}) = (\frac{2}{2}, \frac{4}{2}) = (1, 2)$

Problem 29

Find the midpoint of the segment connecting points $A (6, 8)$ and $B (10, 14)$ .

Solution: The midpoint $M$ is:

$M = (\frac{6 + 10}{2}, \frac{8 + 14}{2}) = (\frac{16}{2}, \frac{22}{2}) = (8, 11)$

Problem 30

The points $A (4, 2)$ and $B (- 2, 6)$ define a line segment. What is the midpoint of this segment?

Solution: The midpoint $M$ is:

$M = (\frac{4 - 2}{2}, \frac{2 + 6}{2}) = (\frac{2}{2}, \frac{8}{2}) = (1, 4)$

By: Mohammad Kabir

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