Perpendicular lines and perpendicular bisectors
Use a negative reciprocal gradient, including horizontal–vertical exceptions, and bisect a segment accurately.
Extended only.
Before you begin
- Find gradients, line equations and midpoints.
- Calculate a signed reciprocal.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What is the negative reciprocal of 4/3?
What you will learn
- Find a perpendicular line through a point.
- Find the equation of a perpendicular bisector.
- Handle horizontal and vertical lines without dividing by zero.
A right angle changes the gradient
For two non-vertical lines with non-zero gradients, perpendicularity means m₁m₂ = −1. Therefore a line of gradient 4/3 has perpendicular gradient −3/4. Both the sign and the reciprocal change. Merely negating 4/3 to −4/3 does not usually make a right angle.
Use the reciprocal of the fully simplified gradient and change its sign. A gradient −2 has perpendicular gradient 1/2. The product check (−2)(1/2) = −1 confirms the relationship. The rule assumes equal physical units on the two coordinate axes; a visually stretched graph can distort apparent angles.
Pass the perpendicular through the required point
Once the perpendicular gradient is known, write y = m₂x + c and substitute the specified point. For a line perpendicular to y = 2x + 3 through (4, 1), m₂ = −1/2. Then 1 = −2 + c gives c = 3, so the equation is y = −(1/2)x + 3.
A horizontal line has gradient zero, so its perpendicular is vertical. Through (4, 1) that vertical equation is x = 4. A line perpendicular to a vertical line is horizontal, here y = 1. These are geometric exceptions to using a finite negative-reciprocal calculation.
A bisector must also go through the midpoint
The perpendicular bisector of AB is a line at right angles to AB through its midpoint. For A = (−3, −2) and B = (3, 6), AB has gradient 8/6 = 4/3 and midpoint (0, 2). The bisector therefore has gradient −3/4 and equation y = −(3/4)x + 2.
Check both conditions: the gradient product is −1, and the midpoint satisfies the bisector equation. Passing a perpendicular through endpoint A would make a right angle but would not bisect the segment. Every point on the perpendicular bisector is equally distant from the two endpoints; this gives a useful map or construction interpretation.
Pause and explain
A perpendicular bisector must pass through which point?
Worked example
Find the perpendicular bisector of the segment joining (−3, −2) and (3, 6).
Show the worked solution
- The midpoint is ((−3 + 3)/2, (−2 + 6)/2) = (0, 2).
- The segment gradient is (6 + 2)/(3 + 3) = 4/3, so the perpendicular gradient is −3/4.
- Through (0, 2), the intercept is two: y = −(3/4)x + 2.
Answer y = −(3/4)x + 2
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 line perpendicular to y = 2x + 3 through (4, 1).
Give me a hint
Find the negative reciprocal before solving for the intercept.
Compare my reasoning
- The perpendicular gradient is −1/2.
- Use 1 = (−1/2) × 4 + c to get c = 3.
- The line is y = −(1/2)x + 3; its gradient product with the original is −1.
y = −(1/2)x + 3
Look for these in your work
- Used the negative reciprocal.
- Passed the line through the specified point.
Find the perpendicular bisector of the segment from (−2, 1) to (4, 1).
Give me a hint
First decide whether the segment is horizontal or vertical.
Compare my reasoning
- The segment is horizontal because both y-values are one.
- Its midpoint is (1, 1).
- The bisector is the vertical line x = 1; no finite gradient is needed.
x = 1
Look for these in your work
- Recognised the horizontal–vertical exception.
- Used the midpoint rather than an endpoint.
Two radio towers are at A = (−3, −2) and B = (3, 6). Verify that P = (4, −1) lies on their perpendicular bisector and is equally distant from both.
Give me a hint
Test the line equation, then compare squared distances.
Compare my reasoning
- The bisector is y = −(3/4)x + 2; at x = 4 it gives y = −1.
- PA² = (4 + 3)² + (−1 + 2)² = 49 + 1 = 50.
- PB² = (4 − 3)² + (−1 − 6)² = 1 + 49 = 50, so both distances are √50.
P is on the bisector; PA = PB = 5√2 units
Look for these in your work
- Checked both the equation and equal-distance interpretation.
- Compared distances without unnecessary rounding.
Common mistakes
- Changing the sign without taking the reciprocal.
- Passing a bisector through an endpoint.
- Dividing by zero for horizontal or vertical exceptions.
What can you explain now?
What line through (2, −3) is perpendicular to x = 5?
Compare with the explanation
y = −3
The original line is vertical, so its perpendicular is horizontal at the specified point's y-coordinate.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Tomorrow, form a perpendicular bisector from new endpoints and test its gradient and midpoint conditions separately.
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.
Original StudyForA teaching content · AI-assisted checks · Human teacher review pending.
Number contains 19 sequenced lessons; Algebra and graphs contains 21; Coordinate geometry contains 10 teaching lessons; Geometry contains 15. Coordinate geometry and Geometry each have a mixed checkpoint. Nine earlier overviews support selected topics across the syllabus. The other five syllabus areas, full cumulative assessment and human teacher review are not yet complete. Written lesson practice and paper constructions are self-checked, not automatically graded. The checkpoints sample skills and do not save an exam grade or certify mastery.
Check the official syllabus 2025–2027 Open related practice and resources