Midpoints and a missing endpoint
Average each coordinate separately, then reverse the relationship to find an unknown endpoint.
Section Paper 2 · Extended · Non-calculator
Extended only. This is the separate 9–1 qualification 0980. Cambridge intends Core for grades 4–1 and Extended for grades 9–5; the eligible award bands are grades 5–1 for the Core papers and grades 9–3 for the Extended papers.
Before you begin
- Calculate the mean of two signed numbers.
- Solve a simple linear equation.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What is the mean of −3 and 3?
What you will learn
- Calculate a midpoint between two points.
- Recover an endpoint when the midpoint and other endpoint are known.
- Check that a midpoint bisects the segment.
Halfway in both coordinates
The midpoint of A = (x₁, y₁) and B = (x₂, y₂) is M = ((x₁ + x₂)/2, (y₁ + y₂)/2). Average the two horizontal positions and the two vertical positions separately. For (−3, −2) and (3, 6), the midpoint is (0, 2). Both coordinates move halfway through the corresponding change.
This formula uses sums, unlike the distance formula's differences. Writing the two coordinate calculations on separate lines prevents mixing an x-value with a y-value. The midpoint can have fractional coordinates even when both endpoints have integer coordinates.
Reverse an average to find the missing endpoint
If A and M are known, the sum of the endpoint x-coordinates is 2x_M. Hence x_B = 2x_M − x_A; similarly y_B = 2y_M − y_A. For A = (−2, 5) and M = (1, 3), B has x = 2(1) − (−2) = 4 and y = 2(3) − 5 = 1.
Do not average A and M: that finds a point halfway along only half the original segment. Instead, the displacement from A to M must be repeated from M to B. In the example that displacement is (3, −2), so adding it to (1, 3) also gives (4, 1).
Check by position and equal changes
A midpoint must lie between the endpoints along their straight segment. Its x-coordinate is between the endpoint x-values, and its y-coordinate is between their y-values, including equality for horizontal or vertical lines. This is a useful plausibility check but does not alone prove a candidate is the midpoint.
A stronger check compares the changes A to M and M to B. Both horizontal changes must match, and both vertical changes must match. Then the two half-segments have the same direction and length. For the diagram, A to M and M to B each change by (3, 4), so each half has length five.
Pause and explain
A = (−3, −2), B = (3, 6). What is their midpoint?
Worked example
A = (−2, 5) and the midpoint of AB is M = (1, 3). Find B.
Show the worked solution
- The endpoint x-values sum to 2 × 1 = 2, so x_B = 2 − (−2) = 4.
- The endpoint y-values sum to 2 × 3 = 6, so y_B = 6 − 5 = 1.
- Check the midpoint of (−2, 5) and (4, 1): (2/2, 6/2) = (1, 3).
Answer B = (4, 1)
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Find the midpoint of (−4, 1) and (2, 7).
Give me a hint
Average x with x and y with y.
Compare my reasoning
- The x-coordinate is (−4 + 2)/2 = −1.
- The y-coordinate is (1 + 7)/2 = 4.
- Both endpoints are displaced by (3, 3) on either side of the midpoint.
(−1, 4)
Look for these in your work
- Kept each coordinate pair separate.
- Checked the equal half-displacements.
Find B when A = (3, −4) and M = (−1, 2) is the midpoint of AB.
Give me a hint
Double each midpoint coordinate and subtract the known endpoint coordinate.
Compare my reasoning
- x_B = 2(−1) − 3 = −5.
- y_B = 2(2) − (−4) = 8.
- Check: the means of 3 and −5, and of −4 and 8, are −1 and 2.
B = (−5, 8)
Look for these in your work
- Reversed the midpoint relation.
- Checked the resulting endpoint.
A cable joins supports at (1, 2) and (8, 5). A sensor is installed halfway along the straight cable. Give its coordinates and explain why rounding them to integers changes its position.
Give me a hint
A halfway location can have half-unit coordinates.
Compare my reasoning
- Its x-coordinate is (1 + 8)/2 = 4.5.
- Its y-coordinate is (2 + 5)/2 = 3.5.
- Rounding either coordinate moves the sensor away from the exact midpoint.
(4.5, 3.5)
Look for these in your work
- Retained fractional coordinates.
- Distinguished exact location from a rounded approximation.
Common mistakes
- Using differences rather than means.
- Averaging the known endpoint and midpoint to find the other endpoint.
- Rounding a half-unit coordinate unnecessarily.
What can you explain now?
Find the midpoint of (0, −6) and (0, 4).
Compare with the explanation
(0, −1)
The horizontal average is zero and the vertical average is (−6 + 4)/2 = −1.
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Make it stick
Make up endpoints with negative and fractional coordinates, calculate their midpoint, then recover an endpoint from the other two points.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.