Midpoints

Video lesson · from 7:572 worked examples

One lesson video covers all of The Straight Line, so it opens at 7:57 for this topic — not from the beginning.

Theory

The midpoint is the exact centre point of a straight line connecting two points.

The midpoint of $(x_1, y_1)$ and $(x_2, y_2)$ is found by averaging the $x$ and $y$ coordinates:

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

⚠️ Common Examiner Traps

Add then divide, do not subtract: the midpoint averages the coordinates. Using the difference gives the length, not the middle.
Working backwards is common: given one endpoint and the midpoint, use $x_2 = 2x_m - x_1$ . Guessing rarely works with negatives.
Careful with negatives: the average of $-3$ and $7$ is $2$ . Write the addition down rather than doing it mentally.
Answer as coordinates: a midpoint is a point, so give it as a pair.

Worked examples

Example 1

Calculate the midpoint of $(1, -4)$ and $(7, 8)$ .

M = \left(\frac{1+7}{2}, \frac{-4+8}{2}\right)

M = \left(\frac{8}{2}, \frac{4}{2}\right)

M = (4, 2)

Example 2

In the diagram, $A(9, -2)$ lies on the circumference of the circle with centre $C(17, 12)$ , and the line $AB$ is the diameter of the circle. Find the co-ordinates of $B$ .

Since $C$ is the centre and $AB$ is the diameter, $C$ is the midpoint of $AB$ .

Let $B = (x, y)$ .

\frac{9+x}{2} = 17 \implies 9+x = 34 \implies x = 25

\frac{-2+y}{2} = 12 \implies -2+y = 24 \implies y = 26

The co-ordinates of $B$ are $(25, 26)$ .