lesson
Construct a perpendicular bisector
1 · Learn
A segment's perpendicular bisector crosses its midpoint at 90°. Every point on this line is equally far from the two endpoints. We can locate two such points with equal-radius compass arcs rather than guessing the midpoint.
Draw segment AB, 6 cm long. Open the compass to 5 cm, which is more than half of AB. With centre A draw arcs above and below AB. WITHOUT changing the opening, repeat from B so the arcs cross at P and Q. Use a ruler to draw the straight line PQ. Label where it crosses AB as M. Leave the arcs visible.
Why does it work? AP = BP because both are compass radii, and AQ = BQ for the same reason. P and Q therefore lie on the equal-distance line. Triangles APQ and BPQ have three corresponding equal sides, so their matching angles are equal. PQ passes through the midpoint and is perpendicular to AB. The whole line, not just M, is the locus of points equally far from A and B.
2 · Worked example
After constructing AB = 6 cm with equal 5 cm arcs, measure AM and MB: both should be 3 cm. A radius of 2 cm would fail: two such circles cannot meet because 2 + 2 < 6. A radius of exactly 3 cm makes them touch only once, so it does not give the two intersection points needed for this method.
3 · Your turn
Construct the perpendicular bisector of an 8 cm segment using equal 5 cm radii. What should the two half-lengths and crossing angle be? Explain why changing the opening between centres is a mistake.
Check your answer
Half-lengths 4 cm and 4 cm; angle 90°. With different radii, an intersection is not equally distant from the two endpoints, so this construction's justification fails.
4 · Apply your learning
Choose two different points on your constructed line and measure their distances to both endpoints. Compare the pairs; small differences can come from drawing or measuring error. Do not treat a few measurements as a proof for every point.
Adult guidance and safety
Demonstrate safe compass handling. Use paper, pencil, ruler and compass; keep construction arcs visible. A protractor can check the result but does not replace a compass construction. A drawing pad can record a sketch, but its freehand lines do not establish compass accuracy.
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Check a calculation · Year 5 onward
Use this only when your teacher or adult allows a calculator for the task. Show your thinking first. It does not replace mental or written arithmetic and does not mark your lesson answer.
Powers, roots and trigonometry · Year 7 onward
Use only when the task allows a calculator. Choose one operation at a time and record your working. Trigonometry here always uses degrees (DEG), not radians. Inverse sine, cosine and tangent return an angle, not a reciprocal. Keep exact fractions and π in answers when requested.
Approximate numerical results use up to 12 significant digits. Inverse sine returns −90° to 90°, inverse cosine 0° to 180°, and inverse tangent −90° to 90°; these are principal values, not every possible solution of a trigonometric equation.
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