Geometric Constructions: Step-by-Step Guide
A geometric construction is a drawing made with only a compass (for drawing circles and arcs) and a straightedge (a ruler with no numbers). These tools let you make exact figures. Basic constructions include copying a segment or angle, cutting a segment or angle in half (bisecting it), and drawing perpendicular lines and parallel lines.
Math Racer skill guide · Geometry · Grade 10
How to solve it, step by step
- Decide what you need to build. Pick the right construction: perpendicular bisector, angle bisector, copy of a segment or angle, a perpendicular line through a point, or a parallel line through a point.\[\text{goal: bisect segment } AB\]
- Set the compass. To bisect segment AB, open the compass wider than half of AB. The exact setting does not matter as long as it is more than half.\[r > \tfrac{1}{2}|AB|\]
- Draw arcs from each endpoint. Put the compass point on A and draw an arc above and below the segment. Keep the same setting, put the compass point on B, and draw arcs that cross the first two.\[\text{arc}_A \cap \text{arc}_B = \{P, Q\}\]
- Connect the crossing points. Use the straightedge to draw a line through the two points where the arcs cross, called P and Q. This line is the perpendicular bisector of AB.\[\overleftrightarrow{PQ} \perp AB,\; PQ \text{ bisects } AB\]
- Explain why it works. P and Q are each the same distance from A and from B, because the compass setting did not change. So both are on the perpendicular bisector, and two points make a line.\[PA = PB \text{ and } QA = QB \Rightarrow PQ \perp AB \text{ at midpoint}\]
Worked example
Using only a compass and straightedge, construct the perpendicular bisector of segment AB and explain why it works.
Step 1: Goal: construct the perpendicular bisector of segment AB.
Step 2: Open the compass to more than half of AB.
Step 3: Put the compass point on A and draw an arc that goes above and below the segment.
Step 4: Keep the same setting. Put the point on B and draw a second arc. It crosses the first arc at two points, P (above) and Q (below).
Step 5: Use the straightedge to draw line PQ.
Step 6: Why it works: PA = PB and QA = QB because the compass setting was the same. So P and Q are both on the perpendicular bisector of AB, and the line through them is the perpendicular bisector.
Check: Line PQ crosses AB at its midpoint, and the angle there is 90 degrees.
Tips and common mistakes
- Do not change the compass setting in the middle of a step unless the construction says to.
- Use a sharp pencil and a sturdy compass. A small slip can cause a big error.
- The perpendicular bisector and the angle bisector are the two key constructions. Many others are built from them.
- To copy an angle, draw an arc from the corner, then copy the arc and the distance across it onto the new ray.
- To make a parallel line, copy an angle along a transversal. Equal corresponding angles mean the lines are parallel.
- Every construction works because of a theorem, like congruent triangles or equal radii. Be ready to explain why.
- Draw arcs lightly and the final lines darker. This keeps the drawing easy to read.
- Do not erase your arcs. They show how you built the figure.