Constructions - Standard Maths

The tools and the rules

A construction is an accurate drawing made with only a pair of compasses and a straight edge (a ruler used just for drawing lines, not for measuring). You are not allowed to use a protractor to guess angles. The point of the method is that the compass keeps distances exactly equal, so the result is precise rather than measured by eye.

Construction: An accurate geometric drawing made with compasses and a straight edge alone. All construction arcs are left on the page as evidence of the method.

Never rub out your arcs

The faint arcs are the proof that you used a valid construction. Marks are given for the correct arcs, so leaving them in is not messy, it is required.

Perpendicular bisector

The perpendicular bisector of a line segment \(AB\) is the line that cuts it exactly in half at a right angle. Every point on it is the same distance from \(A\) as it is from \(B\). To construct it:

Open the compasses to more than \(\tfrac{1}{2}\) of the length \(AB\).

With the point on \(A\), draw an arc above and below the segment.

Keeping the same radius, put the point on \(B\) and draw two more arcs so they cross the first pair.

Draw a straight line through the two crossing points. That line is the perpendicular bisector.

Angle bisector

The angle bisector of an angle cuts it into two equal halves. Every point on it is the same distance from both arms of the angle. To construct the bisector of an angle at vertex \(V\):

Put the compass point on \(V\) and draw an arc that crosses both arms of the angle. Call the crossings \(P\) and \(Q\).

Put the point on \(P\) and draw an arc in the middle of the angle.

Keeping the same radius, put the point on \(Q\) and draw an arc that crosses the previous one.

Draw a straight line from \(V\) through that crossing point. It bisects the angle.

Keep the radius fixed

For both constructions, do not change the compass opening between the matching arcs. If the radius drifts, the crossing points move and the line will be wrong.

A worked example

Worked example

A line segment \(AB\) is \(8\) cm long. Describe how to construct its perpendicular bisector, and state one property of the line you draw.

Set the compasses wider than half of \(8\) cm, so more than \(4\) cm.

Draw arcs above and below from \(A\), then, with the same radius, from \(B\), so the arcs cross at two points.

Rule a straight line through the two crossing points.

The line crosses \(AB\) at its midpoint (\(4\) cm from each end) at a right angle, and every point on it is equidistant from \(A\) and \(B\).

Where this is assessed

Producing accurate arcs and lines is Criterion A (Knowing and understanding). Writing the steps in a clear, ordered way that another student could follow is prime Criterion C (Communicating). Using constructions to solve real problems, such as finding a point equidistant from two places, reaches Criterion D (Applying maths in context).

Check yourself

1. Why must the compasses be opened to more than half the length of AB? +

So the arcs from \(A\) and from \(B\) are wide enough to cross each other on both sides of the segment. If the radius were less than half of \(AB\), the arcs would never meet.

2. Give two properties of the perpendicular bisector of AB. +

It passes through the midpoint of \(AB\) at a right angle, and every point on it is the same distance from \(A\) as from \(B\) (it is the set of points equidistant from the two ends).

3. What does an angle bisector produce, and what is special about points on it? +

It splits the angle into two equal halves. Every point on it is the same perpendicular distance from both arms of the angle.

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