The Circumcenter, Incenter and Centroid of a Triangle
You have discovered that the perpendicular bisectors of the sides of a triangle intersect in a point, the angle bisectors
intersect in a point, and the medians intersect in a point. It this portfolio assignment you will investigate to learn about
some special properties of these points.
The Circumcenter:
Take a piece of paper, cut it into a square measuring 5 ½ in. by 5 ½ in. This is the same size as patty paper which is used
to separate meats or cheese at the deli.
Step 1: Draw a large acute scalene triangle on your square paper.
Step 2: Fold three perpendicular bisectors of the triangle.
Step 3: Using your compass, construct a circle with the
circumcenter as its center and passing through one of the
vertex of the triangle.
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Does the circle pass through the other two vertices?
This circle is called the circumscribed circles of the triangle.
Step 4: Repeat steps 1-3 with an obtuse triangle.
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Why does this construction work?
What did you learn about the circumcenter of a triangle and its distance from each vertex?
This distance from the circumcenter to a _________ is the radius of the circumscribed circle.
The Circumcenter, Incenter and Centroid of a Triangle
The Incenter:
STEP 5: Draw a large acute scalene triangle on your square paper.
STEP 6:
Fold the three angle bisectors of the triangle.
STEP 7:
Using your compass, construct a circle with the incenter as its center and
touching one side of the triangle.
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Does the circle touch the other two sides of the triangle? The
circle should touch each side of the triangle at exactly on point.
This circle is called the inscribed circle of the triangle.
STEP 8: Repeat steps 5- 7 with an obtuse scalene triangle.
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Why does this construction work?
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What did you discover about the incenter of a triangle and its distance from each side?
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This distance from the incenter to any side is the ___________ of the inscribed circle.
The Circumcenter, Incenter and Centroid of a Triangle
The Centroid:
Use your square paper.
STEP 9:
Draw a large acute triangle on your square paper and fold or
draw the medians to find the centroid.
STEP 10: What is special about the centrold?
a) To find out, trace your triangle onto a sheet of heavy construction paper. Also trace the medians and the
centroid onto this triangle. You can do this by placing your square paper on top of the construction paper and
redrawing the triangle and medians. If you do this using slightly more pressure than when drawing normally,
you should be able to see an indetation on the construction paper.)
b) Cut out the triangle and see if you can balance it by placing the length of your pencil under a median. Try this
under each median.
c) Now try balancing the triangle on the eraser tip of your pencil. The triangle should balance on the centroid,
because the centroid of mass or center of gravity of the construction paper triangle.
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Does there appear to be anything else that is special about the centroid?
What does the centroid do to each median? (Describe the median segment)
How do these two segments compare in length? Test your conjecture.
STEP 11: On one of the medians mark the length of the shorter segment on the longer segment.
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What do you observe?
How many times with the shorted segment of a median fit on the longer segment of the same median?
Did your observations match your original conjecture?
Complete the following conjecture based on your investigation.
 The centroid of a triangle divides each median into two segments so that the shorter segment is
_______________ the length of the longer segment.