Find midpoints, distances and perpendicular bisectors
Use coordinate formulas to find a segment's length and midpoint, then construct the line that bisects it at right angles.
Section 7 · Straight-line graphs
Cambridge IGCSE Additional Mathematics 0606, 2028–2030 syllabus, Version 1. Paper 1 is non-calculator; Paper 2 requires a scientific calculator. These original lessons provide self-checked topic practice, not a Cambridge grade or a timed paper. Full-paper assessment and human specialist review are still outstanding.
Before you begin
- Cambridge assumes knowledge of IGCSE Mathematics or an equivalent course. Check the domain before using an inverse or logarithm, and retain exact values where the question asks for them.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which point lies halfway between A and B?
What you will learn
- Find the length and midpoint of a segment from its endpoints
- Find a perpendicular bisector using its midpoint and gradient
- Check that the bisector passes through the midpoint and meets the segment at right angles
Midpoint and length come from coordinate differences
For A(x₁, y₁) and B(x₂, y₂), the midpoint is ((x₁ + x₂)/2, (y₁ + y₂)/2). Average each coordinate separately.
The distance is √((x₂ − x₁)² + (y₂ − y₁)²). Squaring removes the sign of each change, and the result is non-negative.
A perpendicular bisector uses the midpoint and a perpendicular gradient
First find the gradient of AB. For a finite non-zero gradient, the perpendicular gradient is its negative reciprocal.
Use the midpoint and perpendicular gradient in point-gradient form. A horizontal segment has a vertical bisector; a vertical segment has a horizontal bisector.
Check the line contains the midpoint. Its gradient relationship also shows it meets AB at 90°; every point on it is equidistant from A and B.
Pause and explain
A segment has gradient 2/5. What is the gradient of its perpendicular bisector?
Worked example
Find the perpendicular bisector of A(−1, 2) and B(5, 6).
Show the worked solution
- The midpoint is ((−1 + 5)/2, (2 + 6)/2) = (2, 4).
- The segment gradient is (6 − 2)/(5 − (−1)) = 2/3, so the perpendicular gradient is −3/2.
- Use y − 4 = −(3/2)(x − 2), or y = −(3/2)x + 7.
Answer y − 4 = −(3/2)(x − 2), or y = −(3/2)x + 7.
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 and distance between (−1, 2) and (5, 6).
Give me a hint
Average coordinates for the midpoint; use the distance formula for the length.
Compare my reasoning
- Midpoint = (2, 4).
- The coordinate changes are 6 and 4.
- Distance = √(6² + 4²) = √52 = 2√13.
Midpoint (2, 4); distance 2√13 units.
Look for these in your work
- Midpoint coordinates averaged
- Both coordinate differences used
- Exact length simplified
Find the perpendicular bisector of the segment joining (1, −2) and (7, 2).
Give me a hint
Find the midpoint and take the negative reciprocal of the segment gradient.
Compare my reasoning
- Midpoint = (4, 0).
- Segment gradient = 4/6 = 2/3, so the perpendicular gradient is −3/2.
- Use y = −(3/2)(x − 4) = −(3/2)x + 6.
y = −(3/2)x + 6.
Look for these in your work
- Midpoint correct
- Perpendicular gradient correct
- Equation passes through midpoint
Two locations are A(2, 5) and B(8, 1). Find the distance and write the perpendicular bisector equation.
Give me a hint
Use the midpoint for the line and the negative reciprocal of AB's gradient.
Compare my reasoning
- Midpoint is (5, 3); distance is √(6² + (−4)²) = 2√13.
- The segment gradient is −4/6 = −2/3, so the perpendicular gradient is 3/2.
- The bisector is y − 3 = (3/2)(x − 5), or y = (3/2)x − 9/2.
Distance 2√13; bisector y = (3/2)x − 9/2.
Look for these in your work
- Distance calculated
- Gradient sign handled
- Bisector uses the midpoint
Common mistakes
- Averaging x with y instead of averaging matching coordinates
- Forgetting the square root in the distance formula
- Using a negative reciprocal without changing sign
- Finding the perpendicular line through an endpoint instead of the midpoint
What can you explain now?
What is the midpoint of (2, −3) and (8, 5)?
Compare with the explanation
(5, 1)
Average x-coordinates to get 5 and y-coordinates to get 1.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
What two pieces of information determine the perpendicular bisector of a segment?
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.