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Geometric Constructions

A free Geometry lesson from the “Geometry Foundations” unit, with a worked example and practice problems including step-by-step solutions.

A geometric construction uses only an unmarked straightedge (for lines) and a compass (for circles and equal lengths). Because circles guarantee equal radii, constructions produce exact results — not just careful drawings. The classical constructions include copying a segment, copying an angle, bisecting a segment, bisecting an angle, and building an equilateral triangle.

What you'll learn

Why it matters: Drafting, architecture, woodworking, and engineering all rely on construction techniques (jigs, templates, layout tools) that mirror compass-and-straightedge methods to produce equal lengths and angles without measurement error.

Worked example

Problem. Outline the steps to construct the perpendicular bisector of segment AB.

  1. Open the compass to more than half the length of AB.
  2. Draw an arc centered at A and another arc (same radius) centered at B.
  3. Connect the two intersection points of the arcs with a straightedge.
  4. That line is the perpendicular bisector of AB.

Answer: Two arcs of equal radius from each endpoint, then connect the intersections.

Practice problems

1. Which two tools are allowed in classical geometric constructions?

Choices: Ruler and protractor · Compass and straightedge · Compass and protractor · Straightedge and graph paper

Show solution
  1. Warm-up: First identify exactly what the question is asking: Which two tools are allowed in classical geometric constructions?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Constructions use a compass for circles and a straightedge for lines.
  4. Rulers and protractors measure — they would not produce exact results.
  5. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Compass and straightedge

2. A straightedge is different from a ruler because it has no:

Choices: Markings or units · Color · Edge · Length

Show solution
  1. Warm-up: First identify exactly what the question is asking: A straightedge is different from a ruler because it has no:
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. A straightedge draws lines; it does not measure them.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Markings or units

3. To copy a segment, you set the compass to the segment's length and then:

Choices: Draw a circle of that radius from the new endpoint · Use a protractor · Trace it freehand · Estimate with a ruler

Show solution
  1. Warm-up: First identify exactly what the question is asking: To copy a segment, you set the compass to the segment's length and then:
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. The radius equals the segment length; the arc marks the new endpoint.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Draw a circle of that radius from the new endpoint

4. To bisect segment AB with compass and straightedge, the compass radius must be:

Choices: More than half of AB · Exactly half of AB · The full length of AB · Less than half of AB

Show solution
  1. Core Practice: First identify exactly what the question is asking: To bisect segment AB with compass and straightedge, the compass radius must be:
  2. For circle problems, connect the formula or theorem to the given radius, diameter, chord, arc, or center information.
  3. Smaller arcs would not intersect.
  4. Equal arcs > half AB produce two intersections on the perpendicular bisector.
  5. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: More than half of AB

5. After drawing the two intersecting arcs, the next step in bisecting a segment is to:

Choices: Draw a circle through both endpoints · Connect the two arc intersections with a straightedge · Erase the arcs · Measure the angle

Show solution
  1. Core Practice: First identify exactly what the question is asking: After drawing the two intersecting arcs, the next step in bisecting a segment is to:
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. The line through both intersections is perpendicular to the segment AND passes through its midpoint.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Connect the two arc intersections with a straightedge

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