Definition

The perpendicular bisector is a line that is perpendicular to a segment and divides it into two congruent segments.

Constructing the Perpendicular Bisector of a Segment (Interactively!)

To construct the perpendicular bisector of segment AB:

• Step 1. Draw two circles with the same radius and with centers at the endpoints of segment AB. The radius must be long enough for the two circles to intersect.
• Step 2. Mark the points of intersection P and Q of the two circles.
• Step 3. Draw line PQ. This is the perpendicular bisector of segment AB.
• Step 4. Mark the intersection M of line PQ with segment AB. This is the midpoint of segment AB.

You can view these steps in the following applet. Drag points A and B to change the length of the segment. You can also adjust the radius of the circles. Use the Measure button to verify that line PQ is perpendicular to segment AB and that point M is the midpoint of segment AB.

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A Closer Look at the Construction of the Perpendicular Bisector — Why Does It Work?

Look at the following diagram. You can drag the endpoints of segment AB or change the size of the radius of the circle.

• Why are segments PA, PB, QA, and QB congruent?
• What can you say about triangles APQ and BPQ? Why?
• Triangles PMA and PMB are congruent. Why?
• What does this imply about segments AM and MB?
• What is the angle between line PQ and segment AB? Why?
• What can we conclude?

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Answer

Segments PA, PB, QA, and QB are congruent because they are radii of the same or congruent circles. Triangles APQ and BPQ are congruent because all pairs of corresponding sides are congruent or coincident. Triangles PMA and PMB are congruent by Side-Angle-Side; they share the common side PM, angles MPA and MPB are congruent because they are corresponding angles in congruent triangles APQ and BPQ, and sides PA and PB are congruent. Segments AM and MB are congruent because they are corresponding sides of congruent triangles PMA and PMB; thus, M is the midpoint of segment AB. Angles PMA and PMB are also congruent and they form a straight angle, so they must both be right angles; therefore, line PQ and segment AB are perpendicular. Our conclusion is that line PQ is the perpendicular bisector of segment AB.

What Are some Applications of Perpendicular Bisectors?

• See how to use perpendicular bisectors to construct the circumcircle of a triangle.

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References and Credits