Construct a perpendicular bisector
Construct perpendicular bisectors with unchanged compass openings and two retained crossings, including tilted segments and an equal-distance application.
Exam scope and assessment notes
Shared Foundation/Higher Construction covers F4.5A–D: measurement and drawing, triangles and other 2D shapes, scale drawings, and both straight-edge-and-compasses bisectors. These six original lessons are linked to those syllabus rows; the links do not establish complete textbook coverage or specialist approval. Use the named instruments and retain your marks, arcs and joins. A numerical answer or displayed reference alone does not show instrument skill. Physical millimetre work needs a real ruler and a verified object or calibrated print; recheck calibration after printing, resizing or zooming. Measure only diagrams explicitly supplied as accurate for measurement. Virtual drawing units do not certify physical millimetres. Written answers and model comparisons are for self-checking, with no marks, mastery or formal assessment approval. The source is linear 4MA1 Issue 2, checked against November 2026/June 2027 entry context; regional options depend on centre eligibility. Modular qualifications remain separate. Publisher chapter links are navigation aids: the relevant final chapter body and Teaching Hub content remain unverified. Other syllabus sections remain separate.
Before you begin
- Use compasses with an unchanged opening and join constructed points with a straight edge.
- Recognise a midpoint, perpendicular lines and a right angle.
- Interpret a supplied parallel guide and compare distances without assuming a page direction.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
For a segment 14 cm long, which equal opening at both endpoints gives two circle crossings?
What you will learn
- Use equal endpoint radii and two arc crossings to construct a segment’s perpendicular bisector.
- Explain why the construction gives both a midpoint and a right-angle crossing.
- Distinguish valid openings, touching or separated arcs, and a supplied guide’s intersection.
Place your own instruments and retain the joins, rays and arcs. The givens are shown first; your construction stays at the positions you choose.
Worked construction
Draw AB=8 cm and construct its perpendicular bisector with straight edge and compasses using the supplied 5 cm opening at both endpoints. Retain arc crossings on both sides of AB and the joining line. Explain why using a 4 cm opening would not give the same two-point construction. Any ruler or protractor check comes after the compass construction.
Drawing unit: cm. Printed graduations calibrate the virtual drawing; they do not certify physical paper or monitor size.
All submitted joins and arcs stay visible. Select a recorded ruler or protractor to inspect its graduations; other instruments retain their actual footprints. Long lines and rays are cropped to this viewing frame.
Coordinates describe the virtual drawing in cm; they are placement controls, not endpoint readings. Positive angles turn anticlockwise; negative sweeps turn clockwise. Clicking chooses a visible position. It does not move an instrument onto a reference answer.
Read or use my recorded ruler
These readings come from your actual ruler placement. A misplaced ruler keeps its offset; this tool does not report the ideal endpoint reading.
Find intersections of my retained traces
Intersections use the actual arc sweeps and line extents. A full circle or supporting line cannot supply a crossing outside a traced arc, ray or segment.
This is a fixed mathematical comparison. A complete written response records an attempt; a teacher still needs to review the reasoning. Virtual graduations do not establish physical ruler, protractor or compass skill, and no marks or mastery are awarded. Verify physical calibration and recheck it after any print or zoom change.
Use one opening from both endpoints
For distinct endpoints A and B, choose one compass opening greater than half their separation. Draw arcs centred at A and at B with that unchanged opening, extending them on both sides of the segment. Mark their two crossings P and Q, then use a straight edge to draw the full line through P and Q. Retain the arcs so the equal-radius method can be inspected.
A ruler may establish a given base length or calibrate an opening, but do not locate a measured midpoint first and simply draw a protractor perpendicular through it. For the required bisector method, the line is determined by the equal-radius arc crossings. An unmarked straight edge joins those constructed points; its measurement marks do not supply the midpoint.
Explain the midpoint and perpendicular conditions
Each arc crossing is equally far from A and B because the two radii are equal. The segment joining the two crossings is the perpendicular bisector: it cuts AB into equal halves and crosses at a right angle. Explain both parts of that description. A perpendicular through some other point, or a line through the midpoint at a sloping angle, would miss one of the two conditions.
After constructing the line, use a ruler or protractor as a check of the equal halves and right angle. Those checks support your explanation but do not replace the retained compass method. The argument concerns the given segment, whether horizontal or tilted; it is not a rule to draw a vertical line on every page. If the endpoints coincide, no unique segment direction is defined and this construction cannot supply a unique perpendicular bisector.
Check openings and apply the equal-distance line
An opening less than half AB gives separated endpoint circles, so there are no crossings. An opening exactly half AB gives only one touching point, which cannot determine the required line by two intersections. Increase the common opening to obtain two usable crossings rather than inventing a second point. The choice of a drawing window or a compass control range is a tool constraint, not a new geometric law.
The perpendicular bisector is also the line of points equally far from the two endpoints. To locate such a point on an additional supplied guide, first construct the bisector and then intersect it with that guide. Compare a trial point’s distances with compasses if needed. This is a chosen application of the same construction, not completion of a general loci programme or evidence of automatic assessment of the learner’s instrument work.
Sources for this lesson
Pause and explain
What do equal radii from A and B establish about an arc crossing P?
Worked example
Draw AB=8 cm and construct its perpendicular bisector with straight edge and compasses using the supplied 5 cm opening at both endpoints. Retain arc crossings on both sides of AB and the joining line. Explain why using a 4 cm opening would not give the same two-point construction. Any ruler or protractor check comes after the compass construction.
Show the worked solution
- Establish the 8 cm segment AB. With the unchanged 5 cm opening, draw arcs centred at A and B above and below the segment.
- Mark both arc crossings P and Q and join them with a straight edge. Since each crossing is 5 cm from each endpoint, the joining line is the equal-distance perpendicular bisector.
- As checks after construction, the line crosses AB at its midpoint, 4 cm from each endpoint, and at 90°. Keep both arcs rather than replacing them with those measured checks.
- An opening of 4 cm is only half the 8 cm separation. The endpoint circles touch once on AB, so there are not two distinct arc crossings from which to draw the required line.
Answer Construct the full line through the two equal-5 cm arc crossings. It bisects AB into 4 cm halves at 90°. A 4 cm opening gives just one touching point, not the required two-crossing construction.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Construct the perpendicular bisector of the supplied 6 cm horizontal segment AB using a 5 cm opening at both endpoints. Retain arcs on both sides, mark the bisector’s intersection M with AB, and check the two halves and the right angle only after constructing the line.
Give me a hint
Keep one opening, find two crossings and join them before using measurement checks.
Compare my reasoning
- Set the compass opening to 5 cm and draw endpoint arcs centred at A and B on both sides of the 6 cm segment.
- Mark the two arc crossings and join them with a straight edge. Their equal endpoint distances locate the perpendicular bisector; its crossing with AB is M.
- After construction, check AM and MB. Their ideal reference lengths are each 3 cm, and the crossing angle is 90°.
- Retain the arcs and explain that the midpoint and right-angle measurements check the result rather than being the method that located the line.
Construct the joining line of the two equal-5 cm arc crossings. At M the 6 cm segment has two 3 cm halves and a 90° crossing, checked after construction.
Look for these in your work
- Use the same 5 cm opening from both endpoints and keep both crossings.
- Draw the bisector through the crossings before measuring its intersection with AB.
- Check and explain the 3 cm halves and 90° crossing without replacing the method.
On the supplied accurately tilted 6 cm segment AB, independently construct its perpendicular bisector using a 5 cm opening at both endpoints. Retain the arcs on both sides and the joining line. Explain why a page-vertical line is not the construction rule; private coordinates or orientation values are not supplied as a shortcut.
Give me a hint
The compass distances depend on the endpoints, not on the page’s horizontal or vertical directions.
Compare my reasoning
- Use the same 5 cm opening at A and B. Draw arcs on both sides of the tilted segment and mark the two crossings.
- Join those crossings with a straight edge and extend the line through the segment. The equal radii make each crossing equally far from A and B.
- Check that the line passes through the segment’s midpoint and crosses AB at 90°. Its page direction is whatever makes it perpendicular to the tilted AB, not automatically vertical.
- Keep the complete construction visible. A private coordinate equation can verify the reference line, but calculating that equation is not the learner’s required straight-edge-and-compasses method.
Draw the full joining line of the two equal-5 cm arc crossings. It is perpendicular to the tilted AB at its midpoint; no page-vertical rule or coordinate calculation replaces the construction.
Look for these in your work
- Construct from the actual tilted endpoints with an unchanged compass opening.
- Retain both arc crossings and draw their full joining line.
- Explain the midpoint and 90° conditions relative to AB rather than the page.
Two pin centres A and B are 9 cm apart on the supplied template. Using a 7.5 cm opening at both centres, construct the straight line on which a new mark would be equally far from the pins. A supplied guide is parallel to AB and 2 cm above it. Locate the qualifying mark R on that guide, then compare the given offset trial point S with the pin centres and explain why it fails. S is a plotted given point, not a preprinted midpoint or solution.
Give me a hint
Construct the equal-distance bisector first, then intersect it with the supplied guide. Compare trial distances with compasses rather than private coordinate formulas.
Compare my reasoning
- With a 7.5 cm opening unchanged, draw arcs from A and B on both sides of AB. Mark their two crossings and join them with a straight edge.
- This joining line is the perpendicular bisector and contains points equally far from the two pin centres. Intersect it with the guide 2 cm above AB to locate R.
- R lies above the midpoint of the pin segment, so its distances to A and B agree. Use compasses to compare those distances as a check after locating R.
- The supplied offset S is not on the constructed bisector. A compass comparison gives different distances from S to the two pins, so S fails the equal-distance condition. Keep the guide, both arcs and the rejected trial visible.
R is the intersection of the constructed perpendicular bisector with the supplied 2 cm-above guide. It is equally far from A and B; the offset trial S is not and must be rejected.
Look for these in your work
- Construct the two-crossing bisector using the common 7.5 cm opening.
- Locate R by intersecting the constructed line with the supplied guide.
- Explain the equal-distance test and reject S without substituting a private coordinate-only solution.
Common mistakes
- Changing the compass opening between the two endpoint centres.
- Replacing the compass method with a measured midpoint and protractor line, or assuming page-vertical is always perpendicular.
- Accepting one touching point or no circle crossing as if it located a unique bisector line.
What can you explain now?
The two equal endpoint circles touch at only one point. Is that enough for the required two-crossing straight-edge-and-compasses bisector method?
Compare with the explanation
No. Increase the common opening to obtain two distinct crossings.
One point alone does not determine a joining line. Although a midpoint or a perpendicular may be checked separately, those measurements do not provide the requested two-arc construction evidence.
After trying it yourself, choose your next review. This is your self-assessment.
Your review choice appears on Today. Sign in to sync it across devices.
Make it stick
State the equal-radius two-crossing method, the midpoint and right-angle conclusions, and the opening condition. Explain why checks with a ruler or protractor do not replace the required compass construction.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.