Learning goals

• Know the basic tools: straightedge (ruler without measurement) and compass.
• Be able to construct: equilateral triangle, perpendicular bisector, angle bisector, perpendiculars, and copy an angle.
• Follow step-by-step constructions used in school geometry and exams.

Tools and notes

• Compass: for drawing arcs with a given radius.
• Straightedge: for drawing straight lines (do not measure with it unless marked).
• Label points clearly (A, B, C...). Work lightly in pencil so constructions can be erased.
• All constructions use only the compass and straightedge.

1. Construct an equilateral triangle on a given segment AB

1. Draw segment AB.
2. With centre A and radius AB draw an arc.
3. With centre B and radius AB draw an arc that meets the first arc at C.
4. Join AC and BC. Triangle ABC is equilateral (all sides = AB).

2. Perpendicular bisector of segment AB

This gives a line that cuts AB into two equal parts at right angles.

1. Draw segment AB.
2. With centres A and B and radius more than half AB draw two arcs that intersect at two points (one above and one below AB).
3. Join the two intersection points. The line is the perpendicular bisector; it crosses AB at its midpoint and at right angle.

3. Bisect an angle

Draw a ray that divides the angle into two equal angles.

1. Given angle ∠XOA, place compass at O and draw an arc that meets both sides of the angle at P and Q.
2. With centres P and Q and the same radius draw two arcs that intersect at R (inside the angle).
3. Draw OR. OR is the angle bisector: ∠XOR = ∠ROA.

4. Perpendicular from a point to a line

Two cases: point P is on the line or off the line.

a) Point P not on the line l

1. From P draw an arc that cuts the line at two points, A and B.
2. With centres A and B and radius > AB/2 draw two arcs that intersect at Q and R on opposite sides of the line.
3. Join Q and R. This line passes through P and is perpendicular to line l.

b) Point P on the line l

1. From P draw any arc that meets the line at A and B on either side of P (use same radius).
2. With centres A and B draw arcs of equal radius above and below the line; join their intersections to get the perpendicular through P.

5. Copy an angle

1. Given angle ∠X, at a new point O' draw a ray O'Y'.
2. With centre at vertex of original angle draw an arc to meet both sides at A and B. With same radius at O' draw an arc meeting O'Y' at A'.
3. Measure the distance AB with compass. With centre A' draw an arc of same length; where that arc meets the first arc from O' is point B'.
4. Join O'B'. Angle O' = original ∠X.

Simple tips and common mistakes

• Keep compass width fixed when a construction step requires 'same radius'.
• Draw light arcs first; darken final lines after checking intersections.
• Label intersections (P, Q, R...) to avoid confusion with many arcs.
• If arcs do not meet cleanly, increase compass radius slightly and redraw.

Practice exercises

1. Construct an equilateral triangle on a given segment of length 6 cm. (Show steps).
2. Given segment AB = 8 cm, construct its perpendicular bisector and mark midpoint M.
3. Bisect a 60° angle — show construction and check using the equilateral triangle method.
4. At point P on line l, construct the perpendicular and name the new point where it meets the perpendicular as Q. Measure PQ.
5. Copy an angle of about 45° from one paper to another using the copying steps.

Answers: follow the step-by-step constructions above. For (3), bisecting a 60° angle gives two 30° angles.