Circumcenter - GeoGebra Dynamic Worksheet
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Theorem 5.3 Circumcenter Theorem
The Circumcenter of a triangle is equidistant from the vertices of the triangle.
This dynamic GeoGebra worksheet will be used to explore the Circumcenter of a
Triangle and how this Circumcenter relates to other aspects of Geometry.
There are 3 important facts when creating a Circumcenter of a Triangle:
1. Construct a perpendicular bisector for each side of the triangle.
2. Where these 3 perpendicular bisectors meet, is the Circumcenter.
3. The Circumcenter is equal distance from each of the 3 angle vertices.
Lesson Goals:
Understand, identify, and use perpendicular and angle bisectors with triangles
the Circumcenter of a triangle.
Key vocabulary words:
Equidistant-line segments that are equal in distance from a common reference p
Perpendicular bisector-a line, segment, or ray that is perpendicular to a side o
a triangle that passes through the midpoint.
Vertex-two sides meet to form an angle.
Glencoe Geometry Textbook Reference:
Section 5-1: Bisectors, Medians, and Altitudes on page 238
See Theorem 5.3 and Example 1 on page 239
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Circumcenter - GeoGebra Dynamic Worksheet
No.
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Name
Definition
1
Point A
One point of triangle ABC, label of po
2
Point B
One point of triangle ABC, label of po
3
Point C
One point of triangle ABC, label of po
4
Polygon TriangleABC
Polygon A, B, C
4
Segment TriangleSidec
Segment[A, B] of Polygon TriangleABC
4
Segment TriangleSidea
Segment[B, C] of Polygon TriangleABC
4
Segment TriangleSideb
Segment[C, A] of Polygon TriangleABC
5
Point MidpointD
midpoint of A, B
6
Point MidpointE
midpoint of C, A
7
Point MidpointF
midpoint of B, C
8
Line PerpendicularBisectorm
Line through MidpointE perpendicular to
9
Line PerpendicularBisectorL
Line through MidpointD perpendicular to
10
Line PerpendicularBisectorn
Line through MidpointF perpendicular to
11
Point JisCircumcenter
intersection point of PerpendicularBisec
12
Segment DistanceAtoJ
Segment[A, JisCircumcenter]
13
Segment DistanceBtoJ
Segment[B, JisCircumcenter]
14
Segment DistanceCtoJ
Segment[C, JisCircumcenter]
Table 1: List of Point and Segment Definitions
Lesson Essential Question: Prove line segments AJ = BJ = CJ
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Circumcenter - GeoGebra Dynamic Worksheet
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Figure 1: Interactive GeoGebra Object
Circumcenter Proof:
Given: L,M, and N are perpendicular bisectors of triangle ABC sides A, B, and C resp
Formal 2 Column Proof:
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Step
Statements
Reasons
1
L is a perpendicular bisector of side A
Given with definition o
2
M is a perpendicular bisector of side B
Given with definition o
3
N is a perpendicular bisector of side C
Given with definition o
4
Point J lies on perpendicular bisector L
Given with definition o
5
Point J is equidistant from point A and B
Definition of perpendic
6
Line segment AJ = BJ
Definition of Equidista
7
Perpendicular bisector N also includes J
Given, definition of per
intersects the Circumc
8
Line Segment BJ = CJ
Definition of Equidista
9
Then Line segment AJ = BJ, BJ = CJ
Restatement from step
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Circumcenter - GeoGebra Dynamic Worksheet
10
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Thus AJ = CJ
______________ Pro
Table 2: Listing of 2 Column Proof
Assignment- Answer the following questions while using the GeoGebra In
1. Name the property in step 9 of the 2 column proof listed above.
2. By moving points A,B, or C on triangle ABC, does the distances of line seg
AJ, BJ, and CJ change?
Use the GeoGebra Interactive object above to test this out.
3. Are line segments AJ, BJ, and CJ always, sometimes, or never equal?
Use the GeoGebra Interactive object above to test this out.
4. Does changing the lengths of Triangle ABC have an effect on segments A
Use the GeoGebra Interactive object above to test this out.
5. Why is point J the Circumcenter of Triangle ABC?
o Explain:_______________________________________________
o The perpendicular bisectors _______________ at point J.
o The angle vertices are ______________ distance from point J.
Norm Ebsary, October 14, 2006, Created with GeoGebra
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