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Length and Area

Coordinate geometry lets you find the length of a line segment, the midpoint of two points, and the area of shapes drawn on a grid — all from the coordinates, using the distance formula and Pythagoras.

25 min Video by Zeeshan Zamurred Straight Line Graphs

Edexcel Straight Lines — full chapter playlist (Zeeshan Zamurred)Watch the full walkthrough before the notes below.

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What you'll be able to do

Find the distance between two points
Find the midpoint of a line segment
Calculate areas of shapes from coordinates
Combine these in geometry problems

Distance between two points

The distance is the length of the line segment, found with Pythagoras on the differences in coordinates.

d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

The distance formula — Pythagoras in disguise.

(4 - 1)^{2} + (6 - 2)^{2} = 9 + 16

Answer

5

Midpoint

The midpoint is the average of the two coordinates.

M = (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})

Average the

x

- and

y

-coordinates.

Areas from coordinates

For a shape with known vertices, use the lengths (and right angles where present) to find the area — for example, base × height for a triangle once you have the relevant side lengths.

\frac{1}{2} \times 6 \times 4

Answer

12

Tip — Sketch the points first — it reveals which sides are the base and height.

Formula recap

d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

Distance between two points.

M = (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})

Midpoint.

Area_{△} = \frac{1}{2} \times base \times height

Triangle area.

Common mistakes to avoid

Forgetting the square root in the distance formula.

Distance = √(Δx² + Δy²).

Subtracting instead of averaging for the midpoint.

Midpoint averages the coordinates.

Key takeaways

Distance = √((x₂−x₁)² + (y₂−y₁)²).
Midpoint = average of the coordinates.
Use lengths and a sketch to find areas of shapes.

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