A “geometric construction” means drawing figures accurately using only a ruler and compass. Do not measure lengths or angles with a protractor; instead, use the compass to transfer “the same length.” First, we will learn how to construct a perpendicular bisector.
Rules for Geometric Constructions
Only the following two tools may be used in middle-school constructions:
・Ruler: Draw straight lines (do not use it to measure lengths).
・Compass: Draw circles and arcs, and transfer lengths to other locations.
A basic rule is to leave the auxiliary lines drawn during the construction, such as arcs and straight lines, without erasing them.
What Is a Perpendicular Bisector?
The perpendicular bisector of line segment AB is the line that passes through AB perpendicularly and intersects it at its midpoint.
・It divides AB into two equal parts.
・It intersects AB at a right angle.
Every point on the perpendicular bisector is at an equal distance from A and B. This property is important.
Constructing a Perpendicular Bisector
Steps for constructing the perpendicular bisector of line segment AB:
① With A as the center, draw arcs with a radius greater than half of AB.
② With the same radius, draw arcs with B as the center.
③ Let the intersection points of the two arcs be P and Q.
④ Draw line PQ → this is the perpendicular bisector.
Construct the perpendicular bisector of line segment AB. Explain why the construction is correct.
Solution:
Follow steps ①–④ above. Since arcs with the same radius were drawn from A and B, both intersection points P and Q satisfy AP = BP、AQ = BQ. Therefore, P and Q are equidistant from A and B and lie on the perpendicular bisector of AB.
Properties and Uses of Perpendicular Bisectors
Points on the perpendicular bisector are equidistant from points A and B.
・It can be used to find the midpoint of a line segment.
・It can also be used to find all points that are equidistant from two points.
What figure represents all the points that are equidistant from two points A and B?
Solution:
The perpendicular bisector of line segment AB. Every point on the perpendicular bisector is the same distance from A and B.
- Geometric constructions use only a ruler (to draw lines) and a compass (to transfer equal lengths and draw arcs).
- Do not erase the arcs you draw.
- Perpendicular bisector: draw two arcs with the same radius from the endpoints and connect their intersections.
- A point on the perpendicular bisector ⇔ the point is equidistant from the two endpoints.
Practice Problems
- State the steps for constructing the perpendicular bisector of line segment PQ.
- Given three points A, B, and C, how can you construct point X that is equidistant from A and B and also equidistant from B and C?
- Describe how to find the midpoint M of line segment AB by construction.
Answers and Explanations
- Answer: With P and Q as centers, draw arcs with the same radius greater than half of PQ, then connect the two intersection points with a straight line.
Explanation: This is the perpendicular bisector. - Answer: Construct the perpendicular bisectors of AB and BC. Their intersection is X.
Explanation: X is equidistant from A and B and from B and C, so it is the center of the circumcircle of ABC. - Answer: Construct the perpendicular bisector of AB. Its intersection with AB is the midpoint M.
Explanation: A perpendicular bisector bisects AB.