Bisectors of Triangles
section 5-2
Whereas many geometric figures have exactly one center (e.g. circle, square, etc.),
the triangle has four different centers. In this section and the next, we will learn
about those four triangle centers. We will see how to locate them and how they can
be used.
Since a triangle has three sides, it has three perpendicular bisectors. When you
construct these perpendicular bisectors, you will find that they have an interesting
property - they all meet at a single point.
Whenever three or more lines intersect at one point, the lines are said to be
concurrent. The point of concurrency is the point where they intersect. This
point of concurrency for the perpendicular bisectors of the sides of a triangle is
called the circumcenter of the triangle. The circumcenter of a triangle is the
center of its circumscribed circle. A circumscribed circle contains all of the
vertices of the polygon around which it is circumscribed.
The circumcenter can be located inside the triangle, outside the triangle, or
on the triangle.
inside the
on the
outside the
H
problem # 1 - Find HZ and GM in
the diagram.
18.6
KZ, LZ, and MZ are all
perpendicular bisectors
of the sides of
GHJ.
K
Z
L
9.5
19.9
G
problem # 2 - Find the circumcenter of
S(0,4), and O(0,0).
M
J
14.5
RSO with vertices R(-6,0),
y
10
8
6
S(0,4)
4
2
0
-10
-8
-6
R(-6,0)
-4
-2
-2
0
2
O(0,0)
4
6
8
10
x
-4
-6
-8
-10
A triangle has three angles, so it has three angle bisectors. The angle bisectors
of a triangle are also concurrent.
This point of concurrency is the incenter of the triangle. The incenter of a
triangle is the center of its inscribed circle. An inscribed circle is tangent to
the three sides of the triangle. In other words, it touches each side at exactly
one point.
Unlike the circumcenter, the incenter is always inside the triangle.
problem # 3 - JV and KV are angle bisectors of
JKL. Find each measure.
K
a. the distance from V to KL
b. m
VKL
W
7.3
V
J
19
106 L
problem # 4 - A city plans to build a firefighters’ monument in the park between
three streets. Draw a sketch to show where the city should place
the monument so that it is the same distance from all three streets.
Justify your sketch.
Constructions: Circumcenter and Incenter of a triangle
Discuss the methods that you would use to construct these triangle centers.