A midpoint is the point exactly halfway between two points, found by averaging the coordinates. A segment length is the distance between them, found with Pythagoras. Both come from the same two coordinates, and exam questions can ask for both in one go.
What do the two formulas say?
For A(x₁, y₁) and B(x₂, y₂):
- Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2). Average the x-values, then average the y-values.
- Length AB = √((x₂ − x₁)² + (y₂ − y₁)²). Find the two differences, square them, add, then take the square root.
The midpoint gives a pair of coordinates. The length gives one number. Knowing the shape of the answer helps you check that you used the right formula.
How do you work it out, step by step?
- Write the two points with brackets round every negative number.
- For the midpoint, add the x-values and halve; add the y-values and halve.
- For the length, find the x-difference and the y-difference.
- Square each difference (a negative difference gives a positive square), then add.
- Take the square root and simplify the surd or round as asked.
Worked example
A is (−2, 1) and B is (6, 7). Find the midpoint of AB and the length of AB.
Midpoint: x = (−2 + 6)/2 = 4/2 = 2, and y = (1 + 7)/2 = 8/2 = 4. The midpoint is (2, 4).
Differences: x-difference = 6 − (−2) = 8. y-difference = 7 − 1 = 6.
Length: AB = √(8² + 6²) = √(64 + 36) = √100 = 10.
Independent check: the differences 6 and 8 form a 3-4-5 triangle scaled by 2, so the hypotenuse is 10. Also, the midpoint (2, 4) is 5 units from A: its differences from A are 4 and 3, and √(16 + 9) = 5, which is half of 10.
The mistake to watch for
The common slip is to add the differences instead of using Pythagoras.
Mistaken working: length = 8 + 6 = 14.
The student found both differences correctly but forgot to square, add and square-root.
A quick sketch shows why 14 is wrong: walking across 8 and then up 6 is a longer route than the straight line between the points. The straight line must be shorter than 14, and in fact it is 10.
A second slip is forgetting to halve when finding the midpoint, which gives (4, 8) instead of (2, 4).
Check yourself
Try these without a calculator, then open each answer.
1. Find the midpoint and length of the segment joining (1, −3) and (7, 5).
Show answer
Midpoint = ((1 + 7)/2, (−3 + 5)/2) = (4, 1). Differences: 6 and 8. Length = √(36 + 64) = √100 = 10.
2. Find the length of the segment joining (−4, 2) and (1, −10).
Show answer
x-difference = 1 − (−4) = 5. y-difference = −10 − 2 = −12. Length = √(25 + 144) = √169 = 13.
3. M(3, 2) is the midpoint of AB, where A is (1, −4). Find B.
Show answer
The midpoint is halfway, so B is as far from M as A is, in the opposite direction. x: 2 × 3 − 1 = 5. y: 2 × 2 − (−4) = 8. So B is (5, 8). Check: the midpoint of (1, −4) and (5, 8) is (3, 2).
Where this leads next
With midpoints and lengths secure, go back to calculating a gradient from two points if the subtraction still feels shaky, or move on to writing the equation of a line. The coordinate geometry practice set includes mixed questions, and the non-calculator working trainer helps with exact surds and fractions.
If you can recite both formulas but mix them up in longer questions, a teacher in online one-to-one Mathematics tuition can watch you work live and train you to sketch first.