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Midpoint of a Line

FoundationHigherAQAEdexcelOCR

Master midpoint of a line for GCSE Maths with structured, exam-style practice. This Foundation and Higher resource covers finding the midpoint of a line segment and includes a complete mark scheme showing the steps examiners reward. Suitable for AQA, Edexcel and OCR. Free to download as a PDF. The midpoint is the average of the x-coordinates and of the y-coordinates.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

Foundation

WorksheetFoundation · PDFMark SchemeFoundation · PDF

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

The midpoint of a line segment is the point exactly halfway between its two endpoints. You find it by averaging the \(x\) coordinates and averaging the \(y\) coordinates separately.

The formula is \(\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\). Because it is just two means, the working is short, but keeping the coordinates in the right pairs matters.

The method works with negative coordinates without any change. Adding a negative simply reduces the total, so the midpoint of \((-4, 2)\) and \((6, -8)\) is \((1, -3)\), which sits between the two as expected.

Revision notes

The midpoint formula

Add the two \(x\) values and halve, then do the same for the \(y\) values.

For \((2, 5)\) and \((8, 11)\): \(\frac{2+8}{2} = 5\) and \(\frac{5+11}{2} = 8\), so the midpoint is \((5, 8)\).

Working with negatives

Keep the signs when adding. A negative coordinate reduces the sum.

For \((-3, 4)\) and \((7, -2)\): \(\frac{-3+7}{2} = 2\) and \(\frac{4+(-2)}{2} = 1\), giving \((2, 1)\).

Working backwards

If you know the midpoint and one endpoint, you can find the other by doubling the midpoint and subtracting.

If the midpoint of \(AB\) is \((4, 5)\) and \(A\) is \((1, 2)\), then \(B\) is \((2 \times 4 - 1,\ 2 \times 5 - 2) = (7, 8)\).

Key points

  • The midpoint is halfway between two points.
  • Average the x coordinates and the y coordinates separately.
  • Midpoint \(= \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\).
  • Keep the signs when adding negatives.
  • The answer is a coordinate pair.
  • Double the midpoint and subtract to find a missing endpoint.

Worked examples

Example 1

Find the midpoint of \((3, 7)\) and \((9, 15)\).

Working

\[\frac{3+9}{2} = 6\]average the x coordinates
\[\frac{7+15}{2} = 11\]average the y coordinates
\[(6, 11)\]write the midpoint as a coordinate pair

Example 2

Find the midpoint of \((-5, 3)\) and \((1, -9)\).

Working

\[\frac{-5+1}{2} = -2\]average the x coordinates, keeping the signs
\[\frac{3+(-9)}{2} = -3\]average the y coordinates
\[(-2, -3)\]state the midpoint

Example 3

The midpoint of \(AB\) is \((5, 4)\) and \(A\) is \((2, 1)\). Find \(B\).

Working

\[2 \times 5 - 2 = 8\]double the midpoint x and subtract A's x
\[2 \times 4 - 1 = 7\]do the same for the y coordinates
\[B = (8, 7)\]state the coordinates of B

Common mistakes

  • Subtracting instead of adding.

    The midpoint uses the mean, so the coordinates are added and halved.

  • Mixing up the coordinates.

    Average the two x values together and the two y values together, never across.

  • Sign errors with negatives.

    For −5 and 1 the sum is −4, giving a midpoint x of −2.

  • Halving only one coordinate.

    Both the x and the y totals must be divided by 2.

Exam tips

  • Write the formula out before substituting.
  • Keep x values and y values in separate calculations.
  • Sense-check that the midpoint lies between the two points.
  • Give the answer as a coordinate pair in brackets.

Key terms

Midpoint
The point exactly halfway between two others.
Line segment
The part of a line between two endpoints.
Endpoint
One of the two points at the ends of a segment.
Mean
The average found by adding and dividing by how many.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.