# Perpendicular Bisector: Definition, Theorem & Equation

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• 0:53 Creating a…
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• 3:18 Real-World Examples
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Lesson Transcript
Instructor
Elizabeth Often

Elizabeth has taught high school math for over 10 years, and has a master's in secondary math education.

Expert Contributor
Kathryn Boddie

Kathryn earned her Ph.D. in Mathematics from UW-Milwaukee in 2019. She has over 10 years of teaching experience at high school and university level.

How can we draw a triangle that will have two exactly equal length sides? Or what if we need to find the center of a circle that passes through a given set of points? In this lesson, we'll learn about the perpendicular bisector and how useful it can be in geometry!

## Definition of a Perpendicular Bisector

How can an archaeologist determine the size of a plate if only a piece of it has been found? How can a landscaper determine sprinkler placement for the most effective water use? It turns out that a single line, called the perpendicular bisector, can be very useful in both of these problems.

The perpendicular bisector of line segment AB is a line that does two things:

• Cuts the line segment AB into two equal pieces or bisects it
• Makes a right angle with the line segment AB (is perpendicular)

An important property is that every point on the perpendicular bisector is the same distance from point A as it is from point B.

## Creating a Perpendicular Bisector

Although you can easily create a perpendicular bisector using geometry software, to do it on paper, all you need is a straightedge and a compass. First, draw your line segment, AB. Then use your compass to create a circle that has point A as its center, and passes through point B. After drawing this circle, draw a second circle that has point B at its center, and passes through point A. The two circles will intersect at two locations, as shown in the picture. A line drawn through the two points of intersection is the perpendicular bisector of the line segment.

## Math Applications

Most applications of the perpendicular bisector are in geometry theorems, proofs, and constructions. For example, you can use a perpendicular bisector to construct a triangle that has two equal length sides, known as an isosceles triangle. If you construct the perpendicular bisector of the line segment AB, every point on the perpendicular bisector will be the same distance from both point A and point B. To construct your isosceles triangle, you can start from any point on the perpendicular bisector and draw line segments to point A and to point B. The two line segments you have just drawn are guaranteed to be the same length! Additionally, the fact that a given line is a perpendicular bisector of one side of a triangle, and passes through the opposite vertex, is proof that the triangle is isosceles.

You can also use the perpendicular bisector to find the circumcenter of a triangle. This point is the center of a circle that passes through all three corners, or vertices, of a triangle. To find this point, you will construct three perpendicular bisectors, one for each side of the triangle. The point where all three perpendicular bisectors intersect is called the circumcenter. Using this center point, we can draw a circle that passes through all three vertices.

Perpendicular bisectors are also useful in finding the center of a circle. If we are given three points on the circle, point A, point B and point C, then we can draw two line segments, AB and AC. The perpendicular bisectors of these two line segments will always intersect at the center of the circle.

## Real-World Examples

Installing a sprinkler system

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## Archaeological Example

Imagine you are an archaeologist and you just discovered part of a piece of pottery - it looks like it was a plate! Unfortunately, it is only part of a plate and not an entire plate - you really wanted to know the typical sizes of plates used in this ancient society's daily life. However, you have learned about a technique of using perpendicular bisectors to discover the size of the entire plate - even if you only have a portion of the plate! You have measured the straight-line distance between two points on the plate edge, labeled A and B in the figure below, to be 3 centimeters, and the straight-line distance between the two points B and C to be 2 centimeters. You also measured the angle formed by the three points ABC to be 170 degrees. Follow the steps below to try out the technique and see if you can discover the typical size of a plate in this ancient society.

## Materials:

• Paper and pencil
• Ruler (with centimeter measurements)
• Protractor
• Compass

## Steps:

• Recreate the points on the plate on your paper with accurate measurements by constructing an angle of 170 degrees and measuring the appropriate distances to label your points A, B, C.
• Create the perpendicular bisector of the line segment AB. Follow the instructions in the lesson to do so.
• Create the perpendicular bisector of the line segment BC. Follow the instructions in the lesson to do so.
• Label the point where the two perpendicular bisectors intersect. This would be the center of the plate.
• Now find the radius of the plate, the diameter of the plate, and the area of the plate using this information.

## Guide to Solution:

• To create the perpendicular bisector of AB, use the compass with the point on A and the pencil on B to draw a circle. Then, use the compass with the point on B and the pencil on A to draw another circle. Where the two circles intersect (should happen twice) are points on the perpendicular bisector. Use the ruler to connect the points with a line.
• The perpendicular bisector for BC is created the same way.
• The radius can be found by measuring the distance from any of the points A, B, C to the center of the circle found when the perpendicular bisectors intersect.
• The diameter is the radius multiplied by 2.
• The area is given by pi times the radius squared.

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