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High School Geometry: Help and Review13 chapters | 162 lessons

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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.

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!

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.*

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.

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.

*Installing a sprinkler system*

If a landscaper is installing a sprinkler system, he wants to water the grass, but not the surrounding asphalt or gravel. For triangular lots, finding the circumcenter, can help the landscaper decide if just one sprinkler will be an effective choice. In some cases, a single sprinkler can water the entire lot and will not waste water outside the lot. In other cases, the single sprinkler at the center of the circle will water the entire lot, but a lot of water will be wasted outside the lot. The landscaper and client will work together to determine the best solution.

*But,* if the lot is in the shape of an equilateral triangle with three equal sides, the lot can be watered with a single sprinkler with minimal water wasted. The picture below shows how one sprinkler can be used to water a lot in the shape of an equilateral triangle, and with less water wasted.

*How big is the plate?*

When archaeologists find pieces of pottery at a dig site, the perpendicular bisector can help them in reconstructing what the original plates may have looked like. If an archaeologist finds multiple pieces of what appear to be the outside of a single plate, they can use the method described to find the diameter of the plate using the outer edge. If they repeat this process for all the pieces, the archaeologist can determine if all the pieces fit together to form a single plate, or if they really come from multiple plates.

The **perpendicular bisector** is a line that divides a line segment into two equal parts. It also makes a right angle with the line segment. Each point on the perpendicular bisector is the same distance from each of the endpoints of the original line segment. These properties make the perpendicular bisector useful for drawing isosceles triangles, for finding the center of a circle, and for finding the center of a circle that contains all three vertices of a given triangle, called the circumcenter of a triangle.

Perpendicular Bisector |
---|

*Divides a line segment into two equal parts *Forms a 90 degree angle with the line segment * Each point on the perpendicular bisector is the same distance from each endpoint of the line segment |

When you are done, you should be able to:

- Explain what a perpendicular bisector is
- Discuss how to find the perpendicular bisector of a line segment
- List some real-world applications for perpendicular bisectors

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High School Geometry: Help and Review13 chapters | 162 lessons

- Applications of Similar Triangles 6:23
- Triangle Congruence Postulates: SAS, ASA & SSS 6:15
- Congruence Proofs: Corresponding Parts of Congruent Triangles 5:19
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