DWG NO. 01.5 — Lesson 5 of 5

Basic Constructions

Unit 1: Foundations of Geometry · ~15–30 min

Objective

Use a compass and straightedge to copy a segment and an angle, and to construct a perpendicular bisector and an angle bisector.

Why construct?

A construction builds a figure using only two tools: an unmarked straightedge (for drawing straight lines) and a compass (for drawing arcs of a fixed radius) — no ruler, no protractor. That restriction is the point: if a figure can be built from just these postulates, it's exact, not an approximation limited by how carefully you can read a measuring device.

Construction 1 · Copying a segment

Steps

ProblemDraw a starting ray with endpoint C.

1.Open the compass to the exact width of segment AB (point on A, pencil on B).

2.Without changing the compass width, place the point on C and swing an arc that crosses the ray at D.

Segment CD ≅ segment AB

Construction 2 · Copying an angle

Steps

ProblemDraw a ray with endpoint D — this will be the new vertex.

1.With the compass point on the original vertex A, draw an arc crossing both sides of ∠A, at points B and C.

2.Without changing the compass width, draw the same arc from D, crossing the new ray at E.

3.Open the compass to the width of BC. Place the point on E and mark that width, crossing the first arc at F.

4.Draw ray DF.

∠EDF ≅ ∠BAC

Construction 3 · Perpendicular bisector of a segment

Arcs of equal radius from A and B cross above and below the segment; the line through those crossings is the perpendicular bisector, meeting AB at midpoint M.

Steps

ProblemStart with segment AB.

1.Open the compass to more than half of AB. With the point on A, draw arcs above and below the segment.

2.Keeping the exact same width, place the point on B and draw two more arcs, crossing the first two.

3.Draw a line through the two points where the arcs cross.

That line is the perpendicular bisector — it crosses AB at its exact midpoint, at a right angle.

Construction 4 · Angle bisector

Steps

ProblemStart with ∠A.

1.Put the compass point on vertex A and draw an arc crossing both sides, at B and C.

2.Put the compass point on B and draw an arc in the interior of the angle. Repeat from C with the same width, crossing the first arc at D.

3.Draw ray AD.

Ray AD bisects ∠A, so m∠BAD = m∠DAC.

Guided practice

1.After constructing the perpendicular bisector of segment AB, how do you find AB's exact midpoint without measuring?
Show answer
The perpendicular bisector crosses AB exactly at its midpoint by construction — just mark that intersection point.
2.Why must the compass stay open to the exact same width while copying a segment?
Show answer
The arc's radius is what transfers the exact length. Changing the width partway through would copy the wrong distance.
3.When bisecting ∠A, why must the two arcs drawn from B and C use the same compass width?
Show answer
So that point D ends up equidistant from B and C. Any point equidistant from both sides of an angle lies exactly on its bisector.
4.Could you copy an angle using a protractor instead of a compass? Would that count as a true construction?
Show answer
You could measure and redraw the angle with a protractor, but that's only as exact as the protractor's markings. A compass–and–straightedge construction is exact by definition, not just close.
5.What goes wrong if the compass width accidentally changes partway through constructing a perpendicular bisector?
Show answer
The four arcs are no longer equidistant from A and B, so the two crossing points won't lie on the true perpendicular bisector — the resulting line will be tilted or miss the actual midpoint.

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