Ruler and compass constructions
Use ruler and compasses to perform standard constructions: perpendicular bisector of a line segment, perpendicular to a line from or at a given point, and bisecting an angle
What a learner can do afterwards
- Construct the perpendicular bisector of a line segment using compasses and a straight edge
- Construct an angle bisector and verify by measuring both halves
- Construct a perpendicular from a point to a line using compasses only
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A construction is a drawing made with only two tools: a ruler as a straight edge, and a pair of compasses. You never use a protractor. Every construction works because a compass draws every point that sits the same distance from one center point.
To bisect segment AB exactly in half: open your compasses wider than half of AB. Put the point on A and draw an arc above and below the line. Keep the same width, put the point on B, and draw two more arcs that cross the first ones. Draw a straight line through the two crossing points. That line is the perpendicular bisector: it cuts AB in half at a right angle.
To drop a perpendicular from point P to a line: put the compass point on P and draw an arc that crosses the line at two spots. From each of those spots, draw an arc below the line, keeping the same width. Where those two arcs cross, draw a line back to P. That line meets the original line at exactly 90 degrees.
To bisect an angle at point O: put the compass point on O and draw an arc that crosses both arms. Keeping the same width, put the point on each crossing spot and draw two more arcs that meet inside the angle. Draw a line from O through that meeting point. It splits the angle into two equal halves.
Keep your compass width exactly the same for both arcs in one step. If you widen or narrow it partway through, the arcs will not cross where they should, and the construction comes out wrong.
Same compass width, two crossing arcs, then a straight line through the crossing point: that is how you build an exact half or a true right angle without ever using a protractor.
More to read
How to Bisect a Line With a Compass and Straightedge: 9 Steps
Bisecting a line means cutting it into two equal halves with only a compass and a straight edge. Set the compass wider than half the segment, swing matching arcs from each end, and join the two points where the arcs cross. That joining line is the perpendicular bisector.
The construction works because of symmetry. Equal compass widths from both ends force the crossings onto the middle line, and the join meets the segment at 90°. Measuring the two halves confirms the job: each should be exactly half the original.
How to Construct a 60 Degrees Angle Using Compass and Ruler
A 60° angle starts from an equilateral triangle, which a compass builds naturally. Draw a baseline, swing an arc from one end with the compass set to the baseline length, and repeat from the other end. The arcs cross at the third corner, and every angle of the triangle is 60°.
No protractor is needed at any stage. The compass carries the length round, and equal lengths force equal angles. Bisecting the 60° angle afterwards gives 30°, which shows how constructions chain into new angles.
How to Construct a 90 Degrees Angle Using Compass and Ruler
A right angle grows out of a straight line and its perpendicular bisector. Mark two points equally spaced round the centre, then bisect the gap between them with matching arcs. The bisector stands at 90° to the baseline.
Checking uses measurement, not trust. Lay a protractor or a set square on the finished angle and confirm 90°. Each construction in this topic ends the same way: build with compasses, verify by measuring.
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