What instruments are used for geometric constructions?

Protractor (for angle constructions)

A perpendicular bisector of a line segment:

Divides the line into two equal parts at 90°

Passes through one endpoint only

When constructing an angle bisector, which arcs are drawn first?

Arcs from the two points where the first arc meets the angle arms

Two equal arcs from each arm endpoint

To construct a 60° angle, you should:

Draw an equilateral triangle construction

The set of all points satisfying a given condition

A perpendicular line from a point

Which of these describe the locus of points equidistant from two fixed points A and B?

A circle centred midway between A and B

The angle bisector of angle AOB

Which of the following constructions produce a 90° angle?

Perpendicular bisector of a line segment

Angle bisector of a 180° angle

Geometric Constructions

⏱ 8 min✏️ Quiz at the end

Why Constructions?

Geometric constructions are precise drawings made using only a compass and straight edge (ruler). Unlike measuring with a protractor, constructions are mathematically exact — the accuracy comes from the properties of circles and intersecting arcs, not from estimating angles or lengths.

Constructions are important because they demonstrate geometric reasoning and underpin proofs. They also appear in locus problems and real-world applications such as engineering and architecture.

The only tools allowed:

A compass (for drawing arcs and circles)
A straight edge or ruler (for drawing straight lines — not for measuring)
A sharp pencil (for accuracy)

Leave all construction arcs visible — rubbing them out loses the evidence of your method.

Constructing a Perpendicular Bisector

A perpendicular bisector crosses a line segment at its midpoint at exactly 90°.

Steps:

Draw line segment AB.
Open the compass to more than half the length of AB.
Place the compass point on A and draw an arc above and below the line.
Without changing the compass width, place it on B and draw two more arcs crossing the first pair.
Draw a straight line through the two intersection points.

The resulting line crosses AB at its midpoint and is perpendicular to it. Any point on this line is equidistant from A and B — a key locus result.

Constructing an Angle Bisector

An angle bisector divides an angle into two equal halves.

Steps:

Draw the angle at vertex V with two arms.
Place the compass on V and draw an arc that crosses both arms — label these points P and Q.
Place the compass on P and draw an arc in the interior of the angle.
Without changing the width, place the compass on Q and draw another arc crossing the first.
Draw a straight line from V through the intersection point.

This line bisects the angle exactly. Any point on the bisector is equidistant from both arms of the angle.

Constructing a 60° Angle

A 60° angle is constructed by building an equilateral triangle, since all angles in an equilateral triangle are 60°.

Steps:

Draw a straight line and mark a point A on it.
Place the compass on A, set it to any radius r, and draw a large arc that crosses the line at B.
Without changing the radius, place the compass on B and draw an arc that crosses the first arc at point C.
Draw a line from A through C.

The angle CAB = 60°. To construct 30°, bisect this angle. To construct 120°, extend one arm and use the supplementary angle.

Constructing Triangles

You can construct a unique triangle when given enough information.

SSS (three side lengths):

Draw the base AB.
Set compass to the length of the second side and draw an arc from A.
Set compass to the length of the third side and draw an arc from B.
Mark where the arcs cross — this is vertex C.
Draw lines AC and BC.

SAS (two sides and included angle): Use a protractor to mark the angle, then use the compass to mark the two side lengths.

ASA (two angles and included side): Draw the base, then construct or measure the two angles at each end.

Perpendicular from a Point to a Line

To drop a perpendicular from a point P to a line:

Place the compass on P and draw an arc that crosses the line at two points, A and B.
Construct the perpendicular bisector of AB (as above).

The resulting line passes through P and meets the original line at 90°.

The Locus

A locus (plural: loci) is the set of all points that satisfy a given condition.

Condition	Locus
Equidistant from a fixed point	Circle centred at that point
Equidistant from two fixed points A and B	Perpendicular bisector of AB
Equidistant from two lines	Angle bisector of the angle between them
A fixed distance from a line	Two parallel lines (one on each side)

Example: Draw the locus of points that are 3cm from point P — this is a circle with centre P and radius 3cm.

Example: Shade the region that is closer to A than to B — this is the half-plane on the A side of the perpendicular bisector of AB.

Common Mistakes

Changing compass width mid-construction — once set for a step, the width must stay fixed.
Rubbing out arcs — construction marks must remain visible to show your method.
Using a protractor instead of constructions — in construction questions, a protractor is not sufficient; arcs are required.
Inaccurate pencil lines — use a sharp pencil and draw lightly so you can see intersections clearly.
Confusing locus conditions — equidistant from a point gives a circle; equidistant from two points gives a straight line.

Tips and Tricks

Set your compass width carefully each time and hold it steady while drawing arcs.
Make arcs long enough to intersect clearly — short arcs are hard to read.
When constructing a perpendicular bisector, use a compass opening that is clearly more than half the line length so arcs definitely cross above and below.
Label every point you construct as you go — it prevents confusion in multi-step problems.
Practise on spare paper before your final answer to build accuracy and confidence.