Advanced Geometry LT 3.2 – Perpendicular Bisectors
A perpendicular bisector is a line that goes through the midpoint of a segment and is perpendicular to the
segment. A triangle has 3 perpendicular bisectors, one for each side.
Exploring Perpendicular Bisectors
Activity:
1. Use your ruler to find the midpoint of
.
2. Draw a perpendicular line going through the midpoint.
3. Draw a point on your perpendicular bisector and label it P.
4. Measure the distance from that point to both endpoints (PA and PB).
5. What do you notice about the distances?
6. Repeat steps 3-5 with a new point called Q.
Reflect:
1. What do you notice about any point on the perpendicular bisector of a segment?
Exploring Perpendicular Bisectors in Triangles
Activity:
1. Use your ruler to find and mark the 3 midpoints of the sides of
.
2. Use your ruler to draw 3 perpendicular lines going through those midpoints. (How can you be sure it is
perpendicular?)
3. What do you notice about the 3 lines you drew?
4. Repeat steps 1-3 for
and
.
3 or more lines are concurrent if they intersect at the same point. The point of intersection is called the point of
concurrency.
Reflection:
2. When all 3 perpendicular bisectors are drawn in a triangle the point of concurrency is called the circumcenter.
Label the circumcenter P in the triangles above, then use a colored pencil to connect P to the vertices of each
triangle. Measure and compare PA, PB, PC. Then do the same for PD, PE, PF and PG, PH, PI.
3. What do your results from question 2 mean?
4. Does the type of triangle affect the location of the circumcenter? How?
Circumcenter Theorem
The perpendicular bisectors of the sides of a triangle intersect at a point that is equidistant from the vertices of
the triangle.
Practice:
1.
each segment.
a)
b)
2. In
,
your reasoning.
are the perpendicular bisectors of
. Use the given information to find the length of
. Find ZH and HG.
. Find KG and ZJ.
is a right angle and D is the circumcenter of the triangle. If
, what is AB? Explain
3.
4. A company that makes and sells bicycles has its largest stores in three cities. The company wants to build a
new factory that is equidistant from each of the stores. Given a map, how could you identify the location for the
new factory?