The perpendicular bisector of a line segment AB is the line that cuts AB exactly in half and meets it at a right angle. It appears in construction questions, and it is also the answer to “find the points equidistant from A and B”.
This skill opens constructions and loci and is used again when loci are combined later in the module.
What does the perpendicular bisector actually do?
Every point on the perpendicular bisector is the same distance from A as from B. Every point off it is closer to one end. That is why the construction works: two arcs of equal radius, one from each end, cross only at points that are equally far from both.
Think of two friends standing at A and B who want to meet somewhere fair. Any point on the bisector is equally far for both of them.
How do you construct it, step by step?
- Draw the segment AB with a ruler and mark the ends clearly.
- Set the compasses to a radius clearly more than half of AB. Do not change it after this point.
- Place the point on A and draw an arc above and below the line.
- Place the point on B and draw a second pair of arcs with the same radius, crossing the first pair.
- Mark the two crossing points and join them with a ruler. The line you draw is the perpendicular bisector, and it meets AB at its midpoint.
Worked example
Draw a line AB of length 8 cm and construct its perpendicular bisector. Then check the construction.
Step 1, choose a radius: half of AB is 4 cm, so use 5 cm.
Step 2, draw the arcs: with radius 5 cm, draw arcs from A and from B. They cross above and below the line.
Step 3, join the crossing points: draw the line through them. It meets AB at M.
Step 4, check by measuring: AM and MB should each be 4 cm, and the angle at M should be 90°.
Step 5, check by calculation: the crossing points are 3 cm from M. In the right-angled triangle formed, √(4² + 3²) = √25 = 5, which is the radius you used. The numbers agree.
The arcs stay on the page as evidence. Only the final line is drawn firmly.
The mistake to watch for
A common slip is to set a radius that is too small, or to let the compasses close slightly between the two ends.
Mistaken working: radius 3 cm for a segment of 8 cm.
The arcs from A and B reach only 3 cm along the segment, so they never meet.
The correction is to test the radius before drawing long arcs: it must be clearly more than half the segment. Then keep your fingers on the hinge, not the legs, so the width does not drift.
Check yourself
Use a ruler and compasses where helpful, then open each answer.
1. AB is 10 cm. What is the smallest whole-number radius (in cm) you can use for the construction?
Show answer
The radius must be more than half of 10, which is 5. The smallest whole number above 5 is 6 cm.
2. P lies on the perpendicular bisector of AB. PA = 13 cm. How long is PB?
Show answer
Every point on the perpendicular bisector is equidistant from A and B, so PB = 13 cm.
3. AB = 12 cm. A point P on the perpendicular bisector is 8 cm from the midpoint M. Find PA.
Show answer
AM = 6 cm. In the right-angled triangle AMP, PA² = 6² + 8² = 36 + 64 = 100, so PA = 10 cm.
Where this leads next
The same arcs-and-line idea gives the bisector of an angle in constructing an angle bisector. The non-calculator working trainer is useful for checking Pythagoras arithmetic such as the examples above.
Some students can draw the construction correctly but lose the reasoning marks. That is the kind of gap our teachers look for in online one-to-one Mathematics tuition.