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.
Understanding Geometry: Constructing a Perpendicular Bisector
The construction works because each arc is part of a circle. A circle is the set of all points at one fixed distance from its center. When the compass stays at one setting, the two arc intersection points have equal distance from endpoint A and endpoint B.
Call the upper intersection point P and the lower one Q. Segment AP has the same length as BP. Segment AQ has the same length as BQ.
The two triangles formed with base AB, triangle APB and triangle AQB, contain matching side lengths. This creates a strong symmetry around the line through P and Q.
A formal proof uses congruent triangles. Compare triangle APM with triangle BPM, where M is the point where the constructed line meets the original segment. They share side PM.
The compass construction gives AP equal to BP. The line through P and Q acts as a mirror line, so the matching parts on either side of it must agree. Another way to justify the result is to use the equal-distance rule.
Every point on the constructed line is equally far from A and B. The only line with that property is the line that passes through the exact middle of AB at a right angle. This equal-distance rule is often called the locus property of a perpendicular bisector.
The arc intersections must be clear enough to join accurately. A compass opening only slightly greater than half the segment can produce intersections very close to the segment. That makes small drawing errors matter more.
A wider opening usually gives two well-separated intersection points and a more reliable line. The opening must not change between drawing from A and drawing from B.
Even a small change means the arc points no longer represent equal distances. Keep the compass needle fixed firmly at each endpoint, make light arcs first, then draw the final straight line only after both intersections are visible.
Students meet this idea whenever a location must be equally far from two places. If two towns lie at A and B, every possible site that is equally distant from both towns lies on their perpendicular bisector. In a triangle, the perpendicular bisectors of all three sides meet at one point.
That point is equally far from all three vertices, so it is the center of the circle through those vertices. This is useful in geometric design, surveying, map work, and computer graphics.
When learning constructions, focus less on how the finished diagram looks and more on what each mark proves. The arcs are evidence of equal lengths, while the straightedge only connects points already determined by that evidence.
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.