 IGD Perpendicular Bisector

• ## IGD: Perpendicular Bisector

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

### Instructions

• Drag points A and B to change the length of the segment. Use the buttons along the right side to visualize the construction of a perpendicular bisector. (After drawing the circles, you can use the slider to 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 AB.
• The Show Segments button allows you to see four segments that may be helpful when attempting to prove that the construction works (see the Exploration section below).
• Click Start Over to reset the drawing.

### Exploration

To construct the perpendicular bisector of AB:
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.
2. Mark the points of intersection P and Q of the two circles.
3. Draw line PQ. This is the perpendicular bisector of segment AB.
4. Mark the intersection M of line PQ with segment AB. This is the midpoint of segment AB.

Why does this construction work?

Click on the Show Segments button. This will create four line segments, AP, AQ, BP, and BQ.

• Why are segments AP, BP, AQ, and BQ 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 you conclude?

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

### Objectives and Standards

NCTM Standards and Expectations
• Geometry / Measurement
• Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
• High School (9-12)
• Geometry