The idea, step by step
Construct a perpendicular bisector
This line crosses a segment at 90° and divides it into equal parts. Draw arcs from each endpoint using the same radius greater than half the segment length. Join the two arc crossings.
Both crossings are equally far from the endpoints because their distances are the same compass radius. Their joining line is the perpendicular bisector. Changing the compass width between endpoints destroys this argument.
Bisect an angle or construct a triangle
For an angle bisector, draw an arc from the vertex to mark equal-distance points on both arms. Draw equal-radius arcs from these points to meet inside the angle. Join their crossing to the vertex.
For an equilateral triangle on AB, draw circles centred at A and B with radius AB. Join either intersection to A and B. All three sides equal the radius; there is no need to estimate a 60° angle.
A common trap
Any two arcs from the endpoints give the required bisector.
Instead: The radii must be equal and large enough to have two crossings.
See the thinking
Bisect a segment
Construct the perpendicular bisector of a 6 cm segment AB.
- Draw AB accurately and set a radius greater than 3 cm.
- Draw equal-radius arcs from A and B above and below the segment.
- Join their crossings U and V. The line meets AB at its 3 cm midpoint at 90°.
Bisect an angle
An angle is 80°. Describe its bisector construction and find the new angles.
- Mark equal-distance points on both arms with an arc from the vertex.
- Draw equal-radius crossing arcs from those arm points.
- Join the vertex to the crossing. The two angles are equal.
- 80°÷2=40°.
Check your understanding
Is radius 3 cm enough for the usual bisector construction on an 8 cm segment?
Show answer
No.
It must be greater than 4 cm; these circles do not meet.
Your turn
6 questions · 19 marks · Work at your own pace.
Read the questions online
- Draw an 8 cm segment and construct its perpendicular bisector. State a suitable radius.4 marks
- Construct the bisector of a 60° angle. State the new angles.3 marks
- Construct an equilateral triangle with side length 5 cm. Explain why the sides are equal.4 marks
- A point P is equally far from A and B. On which line must P lie?2 marks
- A student changes compass width between the arcs from A and B. Why does the usual proof fail?3 marks
- Two circles have radius 4 cm and centres 8 cm apart. Why are they unsuitable for the usual two-crossing construction?3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Draw an 8 cm segment and construct its perpendicular bisector. State a suitable radius.4 marks
Use the same radius greater than 4 cm at each endpoint.
Join the two crossings.
For example, radius 5 cm; the line crosses at the 4 cm midpoint at 90°. - Construct the bisector of a 60° angle. State the new angles.3 marks
Use equal-distance arm marks and equal-radius crossing arcs.
60°÷2=30°.
30° and 30°. - Construct an equilateral triangle with side length 5 cm. Explain why the sides are equal.4 marks
Draw AB=5 cm.
Draw circles of radius 5 cm centred at A and B.
Join an intersection C to A and B. Both new sides are radii.
Each side is 5 cm. - A point P is equally far from A and B. On which line must P lie?2 marks
The locus of points equally far from the endpoints is their perpendicular bisector.
The perpendicular bisector of AB. - A student changes compass width between the arcs from A and B. Why does the usual proof fail?3 marks
Different radii represent different distances.
The construction needs equal distances to both endpoints.
The crossings are not guaranteed to be equally far from A and B. - Two circles have radius 4 cm and centres 8 cm apart. Why are they unsuitable for the usual two-crossing construction?3 marks
The radius is exactly half the centre distance.
Two distinct crossings are needed to determine the joining line.
They touch at one point only.
Curriculum reference & lesson scope
Australian Curriculum v9 / South Australian Curriculum
Angle reasoning consolidation; enlargements using coordinates/digital geometry; invariant and changing properties; geometric constructions; sequences of steps and decisions; testing/refining construction algorithms; explaining geometric reasoning.
Official curriculum source ↗