Geometry: An Interactive Journey to Mastery

Example 2
Figure 12.1 is a map of two intersecting paths in a park.
At point A stands a statue, and at point B is a fountain.
Mark on this page the exact location Jenny should stand
so that she is the same distance from each of the two paths
and, at the same time, equidistant from points A and B.
In fact, show that there are two different places
Jenny could stand to meet the requirements of
this challenge.
Solution
Jenny should stand at the location where
the angle bisector of the two lines and the
perpendicular bisector of AB intersect.
ͼ௘6HHFigure 12.2௘ͽ
Because there are two angle bisectors for the pair of
lines, there is a second location Jenny could stand.
ͼ௘6HHFigure 12.3௘ͽ
Study Tip
x If you are ever asked to prove that two given
line segments are perpendicular, with at least
one segment bisecting the other, consider
looking for equidistant objects in the problem.
Pitfall
x If points P and Q are each equidistant from
points A and B, then PQ is the perpendicular
bisector of AB. Only the segment AB is
guaranteed to be bisected.

A

B

Figure 12.1

A

B

perpendicular bisector

angle bisector

Here!

Figure 12.2

A

B

perpendicular bisector

angle bisectoranother

Here!

Figure 12.3