Geometric Constructions

Creating precise figures with compass and straightedge

What are Geometric Constructions?

Geometric constructions are drawings made using only a compass and straightedge (unmarked ruler). No measurements are taken — precision comes from the geometric properties themselves.

These techniques have been used since ancient Greece and form the foundation of geometric reasoning.

Essential Tools

Compass

Draws circles and arcs

Straightedge

Draws straight lines

Pencil

Makes marks

Basic Constructions

1. Bisecting a Line Segment

To find the exact midpoint of a line segment AB:

1
Set compass width to more than half the length of AB
2
Draw arc from A — place compass point on A and draw an arc above and below the line
3
Draw arc from B — without changing compass width, place point on B and draw arcs that intersect the first arcs
4
Draw the bisector — connect the two intersection points. This line crosses AB at its midpoint M

2. Bisecting an Angle

To divide angle ABC into two equal parts:

1
Draw an arc from vertex B that crosses both arms of the angle (at points P and Q)
2
Draw arcs from P and Q with equal radius so they intersect at point R
3
Draw line BR — this is the angle bisector

3. Constructing a Perpendicular from a Point

To draw a line perpendicular to line L through external point P:

1
Draw an arc from P that crosses line L at two points (A and B)
2
From A and B, draw arcs below the line that intersect at point Q
3
Draw line PQ — this is perpendicular to L

4. Constructing a 60° Angle

To construct a 60° angle at point A on line L:

1
Draw an arc from A that crosses line L at point B
2
Keep the same compass width and draw an arc from B that intersects the first arc at C
3
Draw line AC — angle BAC = 60°

This works because ABC forms an equilateral triangle!

Interactive Construction Simulator

Try It Yourself

Select a tool and click on the canvas to construct

Click to place points on the canvas

Constructing Special Angles

Angles You Can Construct

60° — Using equilateral triangle method

90° — Perpendicular bisector

30° — Bisect a 60° angle

45° — Bisect a 90° angle

120° — Construct 60° and extend (or two 60° angles)

Any angle can be bisected to get half its measure

Practice Problems

Question 1: What is the first step to bisect a line segment AB?

Show Answer

Set the compass width to more than half the length of AB. This ensures the arcs drawn from A and B will intersect.

Question 2: How would you construct a 30° angle?

Show Answer

First construct a 60° angle using the equilateral triangle method, then bisect that 60° angle to get two 30° angles.

Question 3: Why does the 60° construction work?

Show Answer

When you use the same compass width for both arcs, you create three points that are all equidistant from each other. This forms an equilateral triangle, and all angles in an equilateral triangle are 60°.

Question 4: Describe how to construct a perpendicular bisector of line segment PQ.

Show Answer
  1. Set compass width to more than half of PQ
  2. Draw arcs from P above and below the line
  3. With same width, draw arcs from Q to intersect the first arcs
  4. Connect the two intersection points
  5. This line is perpendicular to PQ and passes through its midpoint

CSEC Exam Tips

  • Always leave your construction arcs visible — they show your method
  • Use a sharp pencil for accuracy
  • Don't change compass width mid-construction unless instructed
  • Label all points clearly
  • Check your work by measuring if time permits
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