Compass and straightedge constructions show how geometric figures can be built using only circles, arcs, and straight lines. This cheat sheet helps students remember the exact steps for common constructions and the reasons those steps work. It is especially useful for proofs, diagram accuracy, and understanding congruence in geometry.
The focus is on precise marks, labels, and logical construction procedures.
Key Facts
- A compass preserves distance, so if its opening is set to , any arc made with that opening marks points the same distance from the center.
- A straightedge draws a line through two points, but it is not used to measure length or mark equal distances.
- The perpendicular bisector of is the line through points where equal-radius arcs from and intersect, and every point on it is equidistant from and .
- To construct the midpoint of , construct the perpendicular bisector of and label its intersection with as , so .
- An angle bisector of divides it into two congruent angles, so .
- To copy from point , draw a ray from , set the compass to , and mark on the ray so .
- To construct a line through perpendicular to line , make two equal-distance marks on from when possible, then construct the perpendicular bisector of the marked segment.
- To construct a line through parallel to line , copy an angle formed by a transversal with at point so the corresponding angles are congruent.
Vocabulary
- Compass
- A tool used to draw circles and arcs and to transfer a fixed distance without measuring.
- Straightedge
- A tool used to draw straight lines through points without using measurement marks.
- Perpendicular bisector
- A line that intersects a segment at its midpoint and forms right angles with the segment.
- Angle bisector
- A ray that divides an angle into two congruent angles.
- Congruent
- Figures, segments, or angles that have the same size and shape, often written using .
- Arc
- Part of a circle drawn with a compass from a fixed center and radius.
Common Mistakes to Avoid
- Changing the compass width during a construction is wrong because equal arcs must use the same radius to prove equal distances.
- Using the straightedge as a ruler is wrong because compass and straightedge constructions do not allow measuring with marked units.
- Drawing construction arcs too short is a problem because the needed intersection points may be unclear or missing.
- Assuming a line is perpendicular just because it looks vertical is wrong because perpendicular lines must be justified by equal arcs, a right angle, or a proven construction.
- Erasing all construction marks is a mistake because arcs, tick marks, and intersection points show why the final figure is valid.
Practice Questions
- 1 Construct the perpendicular bisector of when , then state the length of each half of the segment.
- 2 An angle has measure . If you construct its angle bisector, what is the measure of each smaller angle?
- 3 Copy a segment with onto a ray starting at , and name the new endpoint so that .
- 4 Explain why equal-radius arcs from the endpoints of a segment can be used to construct its perpendicular bisector.
Understanding Compass & Straightedge Constructions
A construction is more than a careful drawing. Each mark creates a fact that can be justified. For example, when two arcs are drawn with the same compass width, their intersection has the same distance from both arc centers.
That fact does not depend on how the page is tilted or how wide the original segment is. Geometry calls the set of all points with a shared property a locus. The perpendicular bisector is a locus of points equally far from two endpoints.
An angle bisector is a locus of points equally far from the two sides of an angle. These ideas explain why construction steps lead to reliable results.
The order of steps matters because later marks depend on earlier ones. Arc intersections must be clear enough to locate exactly. If the compass opening changes by even a small amount between related arcs, the equal-distance argument is lost.
Keep the compass point firmly on the center mark. Use arcs that are large enough to intersect well away from the segment or angle vertex. Very tiny arcs can meet at a shallow angle, making the intersection hard to identify.
A sharp pencil and a straightedge edge without chips make a noticeable difference. Extend lines lightly when needed, then darken only the final constructed line.
Students often meet these constructions inside triangle problems. The perpendicular bisectors of a triangle’s sides meet at one point called the circumcenter. This point is equally distant from all three vertices, so it can be used as the center of a circle through those vertices.
The angle bisectors meet at the incenter. That point is equally distant from the three sides and is the center of a circle that touches each side. Perpendicular lines are used in heights and shortest-distance problems.
A perpendicular from a point to a line gives the shortest path to that line. Parallel lines support angle relationships used in road maps, building plans, tiling patterns, and technical drawings.
A strong construction includes evidence, not just a finished shape. Label original points, arc intersections, and important new points. Leave enough of the arcs visible for someone to see how the result was made.
In a written proof, state the reason behind a claim. Equal compass openings give equal lengths. Points on a perpendicular bisector are equidistant from the endpoints.
Copied angles create equal angle measures. Do not use a ruler scale or a protractor to force the answer into place. Those tools may check a result afterward, but they do not prove the construction.
Focus on the property each step guarantees. That habit makes construction problems much easier to understand and explain.