IGCSE Maths / Graphs of straight lines
The midpoint of a line segment
Edexcel International GCSE Maths (4MA1), chapter 6: Graphs of straight lines, specification point 3.3e-f. Free lesson.
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The midpoint is the point exactly halfway between two endpoints. Find it by averaging the x-coordinates and y-coordinates separately, or work backwards by repeating the step from an endpoint to the midpoint.
How to find a midpoint
A midpoint is halfway along a line segment. Its horizontal coordinate is halfway between the endpoints' x-coordinates, and its vertical coordinate is halfway between their y-coordinates. So you calculate two averages, one for each coordinate. For endpoints (x₁, y₁) and (x₂, y₂), the midpoint is M = ((x₁ + x₂)/2, (y₁ + y₂)/2). Add the two x-values and halve them. Then add the two y-values and halve them.
Finding a midpoint with positive coordinates
Work through the x-values and y-values separately. Then check that each answer lies between the corresponding endpoint values. This confirms that the result is halfway between the two ends.
Taking care with negative coordinates
The same averaging method works when coordinates are negative. Pay close attention to the signs when you add, then check that each midpoint coordinate lies between the matching endpoint coordinates.
Why you add and halve, not subtract
Subtracting the coordinates gives the gaps between the points, not a point halfway between them. A midpoint is an average, so add the matching coordinates and divide by 2. For (2, 3) and (8, 11), subtracting gives (8 − 2, 11 − 3) = (6, 8). This describes the coordinate differences. Averaging gives ((2 + 8)/2, (3 + 11)/2) = (5, 7), the midpoint.
Working backwards to a missing endpoint
The midpoint splits the line segment into two equal steps. To find a missing endpoint, work out the step from the known endpoint to the midpoint, then repeat that same step beyond the midpoint.