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G1.12 Coordinates and the Midpoint of a Line Segment

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01 — Explanation

What do coordinates actually tell me about a point's position?

A coordinate pair $(x, y)$ gives the exact position of a point on a grid called the Cartesian plane. The $x$ -axis runs horizontally (side to side) and the $y$ -axis runs vertically (up and down); where they cross is the origin, $(0,0)$ .

Exam Tip

Remember the order with "along the corridor, then up the stairs" — the $x$ -coordinate (horizontal movement) always comes first, the $y$ -coordinate (vertical movement) always second. Writing $(y,x)$ instead of $(x,y)$ is a very common slip under time pressure.

02 — Explanation

How do I plot a coordinate, including negative ones?

Start at the origin every time. A positive $x$ -coordinate moves right, a negative one moves left; a positive $y$ -coordinate moves up, a negative one moves down.

$B(-2, 5)$ : from the origin, go 2 left, then 5 up.
$C(-4, -1)$ : from the origin, go 4 left, then 1 down.
$D(3, -2)$ : from the origin, go 3 right, then 2 down.

03 — Explanation

How do I read off the coordinates of a point already plotted on a grid?

Reverse the plotting process: from the origin, count how many units left/right you must move to be directly above or below the point (this is the $x$ -coordinate), then count how many units up/down from there to reach it (the $y$ -coordinate).

Common Mistake

Counting grid squares instead of grid lines. Coordinates are read from the lines of the grid, not by counting how many little squares a point sits inside — always count from the origin along the axis lines themselves.

04 — Explanation

How do I find the midpoint of a line segment joining two points?

The midpoint is exactly halfway between two points — found by averaging the $x$ -coordinates and averaging the $y$ -coordinates separately.

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

Example: Find the midpoint between $A(2,3)$ and $B(6,9)$ .

Average the $x$ -coordinates: $\dfrac{2+6}{2}=\dfrac{8}{2}=4$ .
Average the $y$ -coordinates: $\dfrac{3+9}{2}=\dfrac{12}{2}=6$ .
The midpoint is $(4,6)$ .

Exam Tip

The midpoint formula still works perfectly with negative coordinates — just be careful adding negative numbers, e.g. the midpoint of $x_1=-3$ and $x_2=7$ is $\dfrac{-3+7}{2}=\dfrac{4}{2}=2$ , not $\dfrac{-3+7}{2}=5$ .

05 — Explanation

How do I find the coordinates of a fourth point that completes a shape?

Use the symmetry of the shape. For a rectangle or parallelogram, opposite sides are equal and parallel, so the missing vertex's coordinates can be found by matching how far the known vertices shift relative to each other.

Example: A rectangle has vertices $A(1,1)$ , $B(5,1)$ , $C(5,4)$ . Find $D$ .

Going from $A$ to $B$ is a shift of $(+4, 0)$ — this must be mirrored by $D$ to $C$ .
So $D$ is $C$ shifted by $(-4, 0)$ : $D = (5-4, 4) = (1, 4)$ .

Exam Tip

Always sketch the shape roughly on paper first, even without a grid provided — plotting the three known points and visualising where the fourth corner must sit catches an impossible answer (like a "rectangle" where the sides clearly wouldn't be parallel) before you commit to it.

06 — Worked example

Worked Exam-Style Question

Point $P$ has coordinates $(-5, 8)$ and point $Q$ has coordinates $(3, -2)$ . Find the midpoint $M$ of $PQ$ .

Average the $x$ -coordinates: $\dfrac{-5+3}{2} = \dfrac{-2}{2} = -1$ .
Average the $y$ -coordinates: $\dfrac{8+(-2)}{2} = \dfrac{6}{2} = 3$ .
The midpoint is $M(-1, 3)$ .

Exam Tip

After finding a midpoint, a quick sanity check is to see that it lies roughly "between" the two original points on a sketch — here $M(-1,3)$ sits between $P(-5,8)$ and $Q(3,-2)$ in both the $x$ - and $y$ -directions, which confirms the answer is plausible.