Compass and straightedge construction is a classical way to build exact geometric figures using only two ideal tools. The straightedge draws lines through known points, and the compass draws circles centered at known points with radii copied from known distances. This matters because it teaches geometry as a system of logical moves rather than as a collection of measurements.
Many important ideas, such as perpendicular bisectors, angle bisectors, and equilateral triangles, can be built from these simple rules.
In a construction, every new point must come from an intersection of allowed lines and circles. The straightedge is unmarked, so it cannot be used as a ruler, and the compass is used to transfer distances without reading numbers. This restriction forces each step to be justified by congruent radii, circle intersections, or line intersections.
The same ideas connect ancient Greek geometry to modern topics such as proof, symmetry, algebraic numbers, and computer-aided design.
Key Facts
- Allowed straightedge move: draw the unique line through two known points.
- Allowed compass move: draw a circle with a known center through a known point, so r = distance between the two points.
- Circle intersection rule: if two circles have the same radius and centers A and B, their intersection points are each distance r from A and r from B.
- Equilateral triangle construction: draw circles centered at A and B with radius AB, then connect an intersection point C to A and B, giving AB = BC = CA.
- Perpendicular bisector construction: equal-radius arcs from A and B meet at points P and Q, and line PQ is perpendicular to AB and bisects AB.
- Angle bisector construction: equal arcs from the angle sides create points equidistant from the vertex, and a second pair of equal arcs locates a point on the angle bisector.
Vocabulary
- Compass
- A tool used to draw circles and copy distances without using numerical measurement.
- Straightedge
- An unmarked tool used only to draw a straight line through two known points.
- Construction
- A geometric drawing made by a sequence of allowed compass and straightedge moves.
- Locus
- A set of points that all satisfy the same geometric condition, such as being a fixed distance from a center.
- Perpendicular bisector
- A line that crosses a segment at its midpoint and forms right angles with the segment.
Common Mistakes to Avoid
- Measuring with the straightedge is wrong because a construction straightedge has no marks and can only draw lines through known points.
- Changing the compass radius without a reason is wrong because copied distances must come from two known points or a previously constructed length.
- Using a point that has not been constructed is wrong because every point in the diagram must come from an allowed intersection or be given at the start.
- Erasing construction arcs too early is wrong because arcs often show the distance relationships needed to justify why the final figure is correct.
Practice Questions
- 1 Segment AB is 8 cm long. You construct an equilateral triangle on AB using two circles of radius AB. What are the lengths of all three sides of the triangle?
- 2 Segment AB is 12 cm long. You construct its perpendicular bisector using equal-radius arcs from A and B. How far is the midpoint of AB from A, and what angle does the bisector make with AB?
- 3 Explain why the two circle intersections in the perpendicular bisector construction lie on a line that is equally distant from A and B.