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NY Regents Exam - Geometry: Help and Review10 chapters | 127 lessons

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Instructor:
*Beverly Maitland-Frett*

Beverly has taught mathematics at the high school level and has a doctorate in teaching and learning.

This lesson will examine the circumcenter of a triangle. We will discuss the definition, location and demonstrate how to construct and identify the circumcenter.

Let's begin with the following scenario: three communities are located at points *L, M and N* as seen in Figure. 1. If the mayor of the town wants to build a hospital so that each community will travel the same distance to the hospital, where would the exact location of the hospital be?

In order to find the exact location of the hospital, we would have to find the **circumcenter**. The **circumcenter** is the point where all the **perpendicular bisectors** of a triangle meet.

**Perpendicular bisectors** are lines segments, or lines, that bisect another line to form right angles. In other words perpendicular bisectors cut other lines in half and form right angles as they do. We could also say that the perpendicular bisector intersect other lines at their midpoints and form right angles. They all mean the same thing, the thing to remember is that perpendicular means right angle, and bisect means to cut into two equal parts.

We can construct a bisector by folding, using a compass, or by using geometry software. As shown in Figure 2, I constructed the perpendicular bisector of line segment AB using geometer's sketchpad, but you can use a compass and pencil following these steps:

1) Draw a line segment *AB*.

2) Open your compass to a little more than half the line segment.

3) Place your compass at point A and swing an arc above and below the line segment AB.

4) Place your compass at point B and swing an arc above and below the line segment.

5) Where the two arcs meet forms the line for your perpendicular bisector.

6) Draw your perpendicular bisector PQ and labelled your right angle.

Let us revert to our original scenario. Let's connect points *L, M and N* to form a triangle and we are going to draw the perpendicular bisector for each side just like we did for line segment *AB*. If you observe Figure 3, you will see that I constructed all my perpendicular bisectors. Notice that all three perpendicular bisectors meet at point *O*. Therefore, this point is our circumcenter, and would be the location for the hospital. Therefore, once you construct the perpendicular bisector of the three sides of the triangle, the circumcenter will automatically become obvious.

So, how do we know that this point *O* is the same distance from points *L, M and N*? Well, take a look at Figure 4, it's the same triangle, but there are two important differences:

1) The circumcenter is also the center of a circle. This circle goes around the triangle and notice that it touches the three vertices of the triangle, points *L, M and N*. This is part of the reason the point is called the circumcenter, 'circum' means around, like circumference.

2) Line segments *OL, OM and ON* represented by the dotted orange lines are all radii of the same circle. Remember the radius of a circle extends from the center of the circle to any point on the circumference. Therefore people will travel the same distance.

Perhaps someone is wondering, if the circumcenter would vary based on the type of triangle. Well, yes it would. The **circumcenter** changes location depending on the type of triangle.

For a** right** triangle: the circumcenter is on the hypotenuse ( the longest side of the triangle) as in Figure 4.

For an **acute** triangle: the circumcenter is a point inside the triangle, see Figure 5.

For the **obtuse** triangle: the circumcenter is outside of the triangle, see Figure 6.

Now, let us summarize what we have discussed so far.

- The perpendicular bisector is the line or line segment that cuts a line in half and forms right angles as they do.
- The circumcenter is the point where all three perpendicular bisectors of a triangle meet.
- The circumcenter is also the center of a circle that goes around the triangle and touches the three vertices of the triangle, which indicates that all three vertices are the same distance from the circumcenter.
- The location of the circumcenter depends on the type of triangle:

Acute Triangle: Circumcenter is located inside, just like you are inside of your house.

Right Triangle: Circumcenter is located on the hypotenuse, just like you are at standing in the door way of your house.

Obtuse Triangle: Circumcenter is located outside of the triangle, just like you are outside in the yard.

Go ahead, practice some construction and have fun with it.

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NY Regents Exam - Geometry: Help and Review10 chapters | 127 lessons

- Triangles: Definition and Properties 4:30
- Area of Triangles and Rectangles 5:43
- Classifying Triangles by Angles and Sides 5:44
- Perimeter of Triangles and Rectangles 8:54
- Interior and Exterior Angles of Triangles: Definition & Examples 5:25
- How to Identify Similar Triangles 7:23
- Triangle Congruence Postulates: SAS, ASA & SSS 6:15
- Applications of Similar Triangles 6:23
- Congruence Proofs: Corresponding Parts of Congruent Triangles 5:19
- Perpendicular Bisector Theorem: Proof and Example 6:41
- Angle Bisector Theorem: Proof and Example 6:12
- Congruency of Isosceles Triangles: Proving the Theorem 4:51
- Converse of a Statement: Explanation and Example 5:09
- Median, Altitude, and Angle Bisectors of a Triangle 4:50
- Properties of Concurrent Lines in a Triangle 6:17
- Angles and Triangles: Practice Problems 7:43
- Congruency of Right Triangles: Definition of LA and LL Theorems 7:00
- Constructing Triangles: Types of Geometric Construction 5:59
- Constructing the Median of a Triangle 4:47
- The AAS (Angle-Angle-Side) Theorem: Proof and Examples 6:31
- The HA (Hypotenuse Angle) Theorem: Proof, Explanation, & Examples 5:50
- The HL (Hypotenuse Leg) Theorem: Definition, Proof, & Examples 6:19
- AA Similarity Postulate & Theorem 5:56
- Circumcenter: Definition, Formula & Construction
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- Similar Triangles: Definition, Formula & Properties 6:43
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