A perpendicular bisector passes through the exact midpoint of a line segment and meets it at a right angle. To draw one, open compasses to more than half the segment's length, draw arcs from each endpoint so they intersect above and below the line, then join those two points with a ruler. No protractor is needed.
What is a perpendicular bisector?
The word "perpendicular" means at 90° (a right angle). The word "bisector" means "cuts in half". A perpendicular bisector of line segment AB:
- passes through the midpoint of AB, and
- meets AB at exactly 90°.
Every point on the perpendicular bisector is equidistant (the same distance) from A and B. This property is what makes it useful in loci problems.
What equipment do you need?
You need:
- A sharp pencil (construction arcs must be visible but not thick)
- A pair of compasses (the drawing kind, for arcs)
- A ruler (for drawing the final straight line)
- An eraser is helpful but do not remove your construction arcs — they show your method
You do not need a protractor. The 90° angle is produced automatically by the geometric property of the intersecting arcs.
How do you construct a perpendicular bisector step by step?
This method works for any line segment AB.
- Open your compasses to a radius greater than half the length of AB. (More than half ensures the arcs cross both above and below the line. A radius of roughly two-thirds of AB is a good choice.)
- Place the compass point on A. Draw an arc above and an arc below line AB.
- Keep the compass radius the same. Place the compass point on B. Draw another arc above and another arc below line AB.
- Mark the two intersection points (one above AB and one below) where the arcs from A and B cross each other. Label these C and D.
- Draw a straight line through C and D using a ruler. This line is the perpendicular bisector of AB.
- Mark the right angle where the bisector crosses AB, using the right-angle symbol (small square).
How do you check the construction is correct?
Use a ruler to measure from the midpoint to A, and from the midpoint to B — they should be equal. Use a set square (or fold the paper carefully) to confirm the bisector meets AB at 90°. If the two measurements differ, the compass radius was changed between steps 2 and 3. Redo the construction with the radius locked.
| Check | What to look for |
|---|---|
| Equal halves | Distance from midpoint to A = distance from midpoint to B |
| Right angle | Bisector meets AB at 90° (use set square or protractor to verify) |
| Symmetric arcs | Both arcs from A and both arcs from B should be visible and of equal size |
Where is the perpendicular bisector used in maths?
The perpendicular bisector is central to loci problems. The locus of all points equidistant from two fixed points A and B is the perpendicular bisector of AB. Common applications include:
- Finding a midpoint accurately without measuring, when a diagram must be constructed precisely.
- Circumscribing a triangle: the perpendicular bisectors of all three sides of a triangle meet at the circumcentre, the centre of the circle that passes through all three vertices.
- Navigation problems: finding a position that is equidistant from two fixed points (e.g. two radio masts).
What common errors should you avoid?
- Changing the compass radius between the two endpoints. This is the most common mistake; the arcs from A and B must have the same radius to intersect correctly.
- Setting the radius to exactly half the segment. If the radius equals exactly half AB, the arcs just touch at the midpoint and cannot form two distinct intersection points above and below. Use a radius that is clearly more than half.
- Rubbing out the arcs. Examiners want to see your construction lines as evidence of the correct method.
- Drawing the bisector only between C and D. Extend the line beyond both intersection points so it clearly crosses AB.
Frequently asked questions
Do I need to use a specific compass radius?
Any radius greater than half of AB will work. The arcs just need to intersect clearly both above and below the line, so avoid a radius that is only slightly more than half. A radius of about 60–75% of the total length usually gives well-spaced intersection points.
Can I construct a perpendicular bisector without compasses?
Not accurately. Without compasses, you could try folding the paper to bring A onto B, which gives a rough perpendicular bisector along the fold line, but this is not an acceptable geometric construction in an exam context. The compasses method is the required technique.
What is the difference between a perpendicular bisector and an angle bisector?
A perpendicular bisector divides a line segment into two equal halves at 90°. An angle bisector divides an angle into two equal angles. They are different constructions used for different purposes, though both use compasses and a ruler and both leave important arc marks on the diagram.
How does the perpendicular bisector help find the circumcentre of a triangle?
Draw the perpendicular bisector of any two sides of the triangle. The point where these two bisectors cross is the circumcentre — equidistant from all three vertices. You can then draw the circumscribed circle with the compass set to that distance. This is a beautiful result that connects the construction to circle geometry.
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