The idea, step by step
Inputs, steps and conditions
An algorithm specifies the starting objects and a sequence of unambiguous actions. For a perpendicular bisector, begin with a segment AB whose endpoints are distinct and choose one compass radius greater than half AB. Use that same radius from both endpoints.
The equal-radius arcs intersect twice. Joining their intersections gives points equidistant from A and B, so the joining line is the perpendicular bisector. A step list should explain why it works, not merely describe a drawing.
Test the edge cases
A radius smaller than half AB produces no intersection; exactly half gives only one. These choices cannot generate the required two-point construction. Distinguish a logical condition from drawing imprecision.
Digital geometry can help test many lengths and positions. Tests support error finding, but the equidistance/congruence argument supplies the proof that the construction works generally.
A common trap
Using a different compass radius from each endpoint.
Instead: Equal radii make both intersections equidistant from A and B. Unequal radii generally give a different line.
See the thinking
Construct a midpoint
Explain how to construct the midpoint of AB using equal compass radii.
- Choose radius greater than half AB.
- Draw arcs from A and B with that radius, meeting at C and D.
- Draw CD; its intersection M with AB is the midpoint.
- CA=CB and DA=DB place C and D on the perpendicular bisector.
Choose a valid radius
AB is 10 cm long. A student selects compass radius 4 cm. Explain failure and give a working radius.
- Each circle reaches only 4 cm toward the other centre.
- The total reach 8 cm is less than 10 cm, so the circles do not meet.
- Any radius greater than 5 cm, such as 6 cm, works in exact geometry.
Check your understanding
For a segment 8 cm long, why is radius 4 cm unsuitable for the two-intersection construction?
Show answer
It produces only one intersection.
The radius must exceed half the segment length.
Your turn
6 questions · 16 marks · Work at your own pace.
Read the questions online
- Draw a segment AB of length 6 cm. Use ruler and compass to construct its perpendicular bisector. Leave your construction arcs visible.3 marks
- A segment is 12 cm long. Which radii 4, 6 and 8 cm produce two equal-circle intersections?2 marks
- Explain why each arc intersection is equidistant from A and B in this construction.2 marks
- Give a coordinate check for the perpendicular bisector of A=(−2, 1), B=(6, 1).3 marks
- A student measures the midpoint with a ruler and then draws an approximate perpendicular. Is this the same as proving a compass construction? Explain.3 marks
- List one input condition and one output condition for an algorithm constructing a square from side AB.3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Draw a segment AB of length 6 cm. Use ruler and compass to construct its perpendicular bisector. Leave your construction arcs visible.3 marks
Use a radius greater than 3 cm from both endpoints.
Join both arc intersections.
The line through the two equal-radius arc intersections bisects AB at 90°. - A segment is 12 cm long. Which radii 4, 6 and 8 cm produce two equal-circle intersections?2 marks
The radius must be greater than 6 cm.
Only 8 cm. - Explain why each arc intersection is equidistant from A and B in this construction.2 marks
This is how the two circles are constructed.
Both distances equal the unchanged compass radius. - Give a coordinate check for the perpendicular bisector of A=(−2, 1), B=(6, 1).3 marks
The midpoint is(2, 1).
AB is horizontal, so its perpendicular bisector is vertical.
x=2 - A student measures the midpoint with a ruler and then draws an approximate perpendicular. Is this the same as proving a compass construction? Explain.3 marks
Measurement can approximate the result.
A construction proof uses exact geometric relationships.
No. - List one input condition and one output condition for an algorithm constructing a square from side AB.3 marks
A zero-length segment cannot form a square.
The output conditions must define the required shape.
Input: A and B are distinct; output: four equal sides and four right angles.
Curriculum reference & lesson scope
Australian Curriculum v9 and SA Curriculum
Angle and similarity reasoning; congruent/similar triangles in proofs; chords, radii and tangents; angles at the centre and circumference; circle theorems and their applications; connected deductive arguments; construction/spatial algorithms; testing and justifying solutions.
Official curriculum source ↗