A perpendicular bisector is a line that cuts a segment into two equal parts and meets it at a right angle. Constructing one with only a compass and straightedge is a classic geometry skill because it uses distances rather than measurement marks. The method is accurate because it depends on circles and equal radii, not on guessing the midpoint.
It is used in proofs, map problems, triangle constructions, and design layouts.
To construct the perpendicular bisector of segment AB, set the compass wider than half the length of AB and draw arcs from A and from B. The arcs intersect at two points, one above the segment and one below it, because those points are the same distance from A and B. Drawing a straight line through the two intersection points creates the perpendicular bisector.
This line crosses AB at its midpoint and forms two 90 degree angles with AB.
Key Facts
- A perpendicular bisector divides a segment into two congruent parts at a 90 degree angle.
- Use the same compass width from both endpoints A and B so the arcs represent equal distances.
- The compass radius must be greater than half the length of AB for the arcs to intersect in two points.
- If P is on the perpendicular bisector of AB, then PA = PB.
- If PA = PB, then P lies on the perpendicular bisector of AB.
- At the midpoint M of AB, AM = MB and the bisector line satisfies bisector ⊥ AB.
Vocabulary
- Perpendicular bisector
- A line, ray, or segment that passes through the midpoint of another segment and forms right angles with it.
- Compass
- A geometry tool used to draw circles or arcs with a fixed radius.
- Straightedge
- A tool used to draw straight lines without using measurement marks.
- Midpoint
- The point on a segment that is the same distance from both endpoints.
- Equidistant
- Equidistant means being the same distance from two or more points or objects.
Common Mistakes to Avoid
- Changing the compass width between endpoints is wrong because the arcs no longer represent equal distances from A and B.
- Using a compass radius less than half of AB is wrong because the arcs will not intersect in two points needed to define the bisector.
- Connecting an endpoint to an arc intersection is wrong because the perpendicular bisector is the line through the two arc intersection points, not through A or B.
- Marking the midpoint by sight is wrong because the construction is meant to prove the midpoint using equal-radius arcs, not estimate it visually.
Practice Questions
- 1 Segment AB is 10 cm long. What is the smallest compass radius that will create two arc intersections, and why must the actual radius be larger than this value?
- 2 Segment AB has endpoints A(2, 3) and B(8, 3). Find the midpoint M and write the equation of the perpendicular bisector.
- 3 Explain why any point where the two compass arcs intersect must lie on the perpendicular bisector of segment AB.