Topics
3 Coordinate geometry
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Stage 5.1-3

3.01 The midpoint of a segment

Lesson
Worksheet
Practice
Lesson

Midpoint between two points

Exploration

Explore this applet demonstrating the midpoint between two points.

What connections exist between the endpoints of a line segment and the midpoint?

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The coordinates of the midpoint of a segment are half the sum of the x and y-coordinates of the end points.

The midpoint of a line segment is a point exactly halfway along the segment. That is, the distance from the midpoint to both of the endpoints is the same.

The midpoint of any two points has coordinates that are exactly halfway between the x-values and halfway between the y-values. This means we can find the average of the two given x-coordinates to find the x-coordinate of the midpoint, and likewise the average of the two y-coordinates will give us the y-coordinate of the midpoint.

So for points A \left(x_1,y_1\right) and B \left(x_2,y_2\right) the midpoint will be: M \left(\dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2} \right).

Think of it as averaging the x and y-values of the endpoints.

By convention, we work from left to right so the point with the lower x-value is generally considered to be the point corresponding to (x_1,y_1), but obviously if we went from right to left we should still end up with the same midpoint.

Examples

Example 1

M is the midpoint of A(-8,-4) and B(2,8). Find the coordinates of M.

-8
-7
-6
-5
-4
-3
-2
-1
1
2
x
-4
-3
-2
-1
1
2
3
4
5
6
7
8
y
Worked Solution
Create a strategy

Use the midpoint formula.

Apply the idea
\displaystyle M\displaystyle =\displaystyle \left(\dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2} \right)Use the midpoint formula
\displaystyle =\displaystyle \left(\dfrac{-8+2}{2}, \dfrac{-4+8}{2} \right)Susbtitute the coordinates
\displaystyle =\displaystyle \left(\dfrac{-6}{2}, \dfrac{4}{2} \right)Evaluate the addition
\displaystyle =\displaystyle \left(-3,2 \right)Evaluate
Idea summary

For points A \left(x_1,y_1\right) and B \left(x_2,y_2\right) the midpoint will be:

\displaystyle M\left(\dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2} \right)
\bm{(x_1,y_1)}
are the coordinates of the end point
\bm{(x_2,y_2)}
are the coordinates of the other end point

Find an endpoint given the midpoint

What if we are given the midpoint of a segment, and one endpoints points of the segment? How can we reverse our steps above to find the other endpoint?

Examples

Example 2

If the midpoint of A(x,y) and B(16,7) is M(10,2), what are the coordinates of A?

Worked Solution
Create a strategy

Use the midpoint formula for each coordinate separately.

Apply the idea

For the x-coordinate of A:

\displaystyle \dfrac{x_1+x_2}2\displaystyle =\displaystyle xUse the formula for the x-coordinate
\displaystyle \dfrac{x+16}2\displaystyle =\displaystyle 10Substitute the values
\displaystyle x+16\displaystyle =\displaystyle 20Multiply both sides by 2
\displaystyle x\displaystyle =\displaystyle 20-16Subtract 16 from both sides
\displaystyle x\displaystyle =\displaystyle 4Evaluate

For the y-coordinate of A:

\displaystyle \dfrac{y_1+y_2}2\displaystyle =\displaystyle yUse the formula for the y-coordinate
\displaystyle \dfrac{y+7}2\displaystyle =\displaystyle 2Substitute the values
\displaystyle y+7\displaystyle =\displaystyle 4Multiply both sides by 2
\displaystyle y+7-7\displaystyle =\displaystyle 4-7Subtract 7 from both sides
\displaystyle y\displaystyle =\displaystyle -3Evaluate

The coordinates of A are A(4,-3).

Idea summary

To find the missing endpoint, equate the average of each x and y-value of the endpoints to each x and y-value of the given midpoint. Then solve for the unknown coordinates.

Outcomes

MA5.3-8NA

uses formulas to find midpoint, gradient and distance on the Cartesian plane, and applies standard forms of the equation of a straight line

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