Constructing the Perpendicular Bisector of a Line Segment

1. What does 'bisect' and 'perpendicular' mean?

Think about how two friends share a loaf of bread from the roadside bakery at Madina. A fair cut goes exactly through the middle, so each friend gets the same amount. In geometry we do the same with a line segment. To bisect a line segment means to cut it into two equal parts. The point where it is cut is the midpoint.

Two lines are perpendicular when they meet at a right angle (90°), like the corner of your exercise book or the bottom edge of a window and its side. A line that does both jobs, cutting a segment in half and crossing it at 90°, is called the perpendicular bisector of that segment.

You could find the midpoint by measuring with a ruler and halving. But the NaCCA syllabus wants you to do it with only a pair of compasses and a ruler, with no measuring and no protractor. This is the way builders and surveyors worked for thousands of years, and the compass method is more accurate than guessing.

2. Constructing the perpendicular bisector

Here is the whole method. Use a sharp pencil, because a thick arc can hide a 1 mm error. Step through the animation slowly and match each step with the list below.

1. 1Draw the line segment AB (for example 8 cm) using your ruler.
2. 2Open your compasses to more than half of AB. Put the point on A and draw one arc above AB and one arc below AB.
3. 3Keep the same compass opening. Put the point on B and draw two more arcs that cut the first two arcs. Label the crossing points P (above) and Q (below).
4. 4Use the ruler to join P and Q. The line PQ cuts AB at M. PQ is the perpendicular bisector of AB.

▶ Notice that the same compass opening is used from A and from B. That is why P and Q are the same distance from both ends.

Why does it work? The arcs from A and B have equal radii, so P is the same distance from A as it is from B. The same is true for Q. Any point that is equally far from A and B must lie on the perpendicular bisector, so the line through P and Q must be it.

3. Bisecting lines of given lengths

The curriculum asks you to draw and bisect lines of given lengths, such as AB = 8 cm and AB = 5.5 cm. First draw the segment accurately. Place the 0 mark of the ruler exactly on A, and mark B at the correct reading. Then follow the four steps above.

After constructing, you should be able to predict what you will measure: for AB = 5.5 cm the midpoint M is 2.75 cm from A. This is a good check on your accuracy. If your ruler reading shows 2.7 cm or 2.8 cm, you are within the normal tolerance.

A very useful fact: every point on the perpendicular bisector is equidistant from A and B. If X is anywhere on PQ, then XA = XB. A triangle XAB with XA = XB is an isosceles triangle.

4. Using perpendicular bisectors in problems

Once you know the construction, you can use it to solve practical problems. Here are the most common BECE-style situations.

• Finding a midpoint of a segment without measuring.
• Finding a point equidistant from two places, such as a borehole, a bus stop or a mast between two towns.
• Finding the centre of a circle from a chord: the perpendicular bisector of any chord passes through the centre.
• Constructing a right angle at the midpoint of a segment, since the bisector meets the segment at 90°.

In a triangle ABC, if you construct the perpendicular bisectors of two sides, they cross at a point that is the same distance from A, B and C. Draw a circle with that point as the centre and it passes through all three corners. This is how a carpenter finds the centre of a round table top that has been damaged.

Worked examples

Try each one yourself first, then open the solution and compare every step.