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Learn Index · Geometry and Spatial Reasoning Grade 9–10

Geometric constructions

Drawing accurate figures with only a compass and straightedge.

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Drawing without measuring

A geometric construction creates exact figures using only:

  • a compass (for circles and arcs, and for copying lengths)
  • a straightedge (for straight lines; no measuring)

The ancient Greek mathematician Euclid built a whole system of geometry on constructions like these over 2,000 years ago.

Construction 1: Copy a segment

  1. Draw a ray with endpoint P.
  2. Set the compass to the length of the original segment AB.
  3. With the point at P, draw an arc crossing the ray at Q.
  4. PQ = AB.

Construction 2: Perpendicular bisector

The perpendicular bisector of AB cuts it in half at 90°.

  1. Set the compass to more than half of AB.
  2. Draw arcs above and below AB from A.
  3. Keeping the same width, draw arcs from B that cross the first arcs.
  4. Join the two crossing points.

Every point on this line is the same distance from A and B.

Construction 3: Angle bisector

The angle bisector splits an angle into two equal parts.

  1. With the compass point on the vertex, draw an arc crossing both arms.
  2. From each crossing point, draw arcs (same width) that cross each other inside the angle.
  3. Draw a ray from the vertex through that crossing.

Construction 4: Perpendicular through a point on a line

  1. From the point P on the line, draw arcs crossing the line on both sides.
  2. From each crossing, draw arcs above the line that meet.
  3. Join P to the meeting point.

Construction 5: Copy an angle

  1. Draw an arc across the original angle and the same arc from the new vertex on a new ray.
  2. Measure the opening between the arc crossings on the original with the compass.
  3. Transfer it to the new arc and draw the second arm.

Construction 6: Equilateral triangle

  1. Draw segment AB.
  2. With width AB, draw an arc from A and an arc from B.
  3. Their intersection C makes triangle ABC equilateral.

Bisecting a 60° angle gives 30°; bisecting a right angle gives 45°.

Why constructions work

Each construction relies on equal radii (every arc from the same compass setting is the same length) and congruent triangles (often SSS). That’s why they’re exact.

Using constructions

  • Finding the centre of a circle (perpendicular bisectors of two chords meet at the centre).
  • Locating a point equidistant from three towns (circumcentre).
  • Laying out right angles in carpentry and landscaping.

Worked examples

Example 1: Which construction finds the midpoint of a segment? Perpendicular bisector.

Example 2: How can you construct a 45° angle? Construct a perpendicular (90°), then bisect it.

Example 3: Why must the compass width stay the same in step 3 of the perpendicular bisector? So both sets of arcs are equal radii and the crossing points are equidistant from A and B.

Common mistakes

  • Changing the compass width partway through a step.
  • Using the ruler’s markings to measure (constructions don’t measure).
  • Arcs too small to intersect.

Try it yourself

  1. Which tool copies a length?
  2. What is special about every point on a perpendicular bisector?
  3. How would you construct a 30° angle?
  4. What do two perpendicular bisectors of chords in a circle meet at?

Answers: 1) the compass 2) It’s equidistant from both endpoints. 3) Construct an equilateral triangle (60°) and bisect an angle. 4) the centre

Quick recap

  • A geometric construction uses only a compass and straightedge.
  • Key constructions: perpendicular bisector, angle bisector, copying segments and angles.
  • They work because of equal radii and congruent triangles.