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3 Coordinate geometry
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Stage 5.1-2

3.01 The midpoint of a segment

Lesson
Worksheet
Practice
Lesson

The midpoint of a segment

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.

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 is the average of the x and y-values of the endpoints of a line segment.

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 y-coordinate of the midpoint, and likewise the average of the two y-coordinates will give us the x-coordinate of the midpoint.

So for points A (x_1, y_1) and B (x_2, y_2) 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.

Examples

Example 1

M is the midpoint of A(5, -6) and B (5,2).

-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
y

Find the coordinates of M.

Worked Solution
Create a strategy

Find the average of the x and y-values of the endpoints.

Apply the idea

For the x-value:

\displaystyle x_m\displaystyle =\displaystyle \dfrac{x_1+x_2}{2}
\displaystyle =\displaystyle \dfrac{5 + 5}{2}Substitute the x-coordinates
\displaystyle =\displaystyle 5Evaluate

For the y-value

\displaystyle y_m\displaystyle =\displaystyle \dfrac{y_1+y_2}{2}
\displaystyle =\displaystyle \dfrac{-6+2}{2}Substitute the y-coordinates
\displaystyle =\displaystyle -2Evaluate

The midpoint is M (5, -2).

Example 2

If the midpoint of A (x, y) and B (10, 3) is M (8, -1).

a

Find the value of x.

Worked Solution
Create a strategy

Use the formula for finding the x-coordinate of the midpoint.

Apply the idea
\displaystyle x_m\displaystyle =\displaystyle \dfrac{x_1+x_2}{2}
\displaystyle 8\displaystyle =\displaystyle \dfrac{x + 10}{2}Substitute the known coordinates
\displaystyle 16\displaystyle =\displaystyle x+10Multiply both sides by 2
\displaystyle x\displaystyle =\displaystyle 6Subtract 10 from both sides
b

Find the value of y.

Worked Solution
Create a strategy

Use the formula for finding the y-coordinate of the midpoint.

Apply the idea
\displaystyle y_m\displaystyle =\displaystyle \dfrac{y_1+y_2}{2}
\displaystyle -1\displaystyle =\displaystyle \dfrac{y + 3}{2}Substitute the known coordinates
\displaystyle -2\displaystyle =\displaystyle y+3Multiply both sides by 2
\displaystyle y\displaystyle =\displaystyle 5Add 2 on both sides
c

What are the coordinates of A.

Worked Solution
Create a strategy

Use the answers found in parts (a) and (b).

Apply the idea

The coordinates of point A is (6,5).

Idea summary

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

Outcomes

MA5.1-6NA

determines the midpoint, gradient and length of an interval, and graphs linear relationships

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