Imagine an archeologist in Cairo, Egypt, found three bones buried 4 meters, 7 meters, and 9 meters apart (to form a triangle)? The likelihood that more bones are in this area is very high. The archeologist wants to dig in an appropriate circle around these bones. If these bones are on the edge of the digging circle, where is the center of the circle? Can you determine how far apart each bone is from the center of the circle? What is this length?
Perpendicular Bisectors
Recall that a perpendicular bisector intersects a line segment at its midpoint and is perpendicular. Let’s analyze this figure.
Perpendicular Bisector Theorem: If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.
In addition to the Perpendicular Bisector Theorem, we also know that its converse is true.
Perpendicular Bisector Theorem Converse: If a point is equidistant from the endpoints of a segment, then the point is on the perpendicular bisector of the segment.
Proof of the Perpendicular Bisector Theorem Converse:
Given:
Prove:
Statement  Reason 

1. 
Given 
2. 
Definition of an isosceles triangle 
3. 
Isosceles Triangle Theorem 
4. Draw point 
Every line segment has exactly one midpoint 
5. 
Definition of a midpoint 
6. 
SAS 
7. 
CPCTC 
8. 
Congruent Supplements Theorem 
9. 
Definition of perpendicular lines 
10. 
Definition of perpendicular bisector 
Two lines intersect at a point. If more than two lines intersect at the same point, it is called a point of concurrency.
Investigation: Constructing the Perpendicular Bisectors of the Sides of a Triangle
Tools Needed: paper, pencil, compass, ruler
1. Draw a scalene triangle.
2. Construct the perpendicular bisector for all three sides.
The three perpendicular bisectors all intersect at the same point, called the circumcenter.
Circumcenter: The point of concurrency for the perpendicular bisectors of the sides of a triangle.
3. Erase the arc marks to leave only the perpendicular bisectors. Put the pointer of your compass on the circumcenter. Open the compass so that the pencil is on one of the vertices. Draw a circle. What happens?
The circumcenter is the center of a circle that passes through all the vertices of the triangle. We say that this circle circumscribes the triangle. This means that the circumcenter is equidistant to the vertices.
Concurrency of Perpendicular Bisectors Theorem: The perpendicular bisectors of the sides of a triangle intersect in a point that is equidistant from the vertices.
If
Determining Unknown Values
1. Find
From the markings, we know that
To find the length of
Applying the Properties of Perpendicular Bisectors
a) Which segments are equal?
b) Find
c) Is
Yes,
Further Exploration
For further exploration, try the following:
 Cut out an acute triangle from a sheet of paper.
 Fold the triangle over one side so that the side is folded in half. Crease.
 Repeat for the other two sides. What do you notice?
The folds (blue dashed lines)are the perpendicular bisectors and cross at the circumcenter.
Archaeology Problem Revisited
The center of the circle will be the circumcenter of the triangle formed by the three bones. Construct the perpendicular bisector of at least two sides to find the circumcenter. After locating the circumcenter, the archeologist can measure the distance from each bone to it, which would be the radius of the circle. This length is approximately 4.7 meters.
Examples
Example 1
Find the value of
By the Perpendicular Bisector Theorem, both segments are equal. Set up and solve an equation.
Example 2
Determine if
2.
Review
m is the perpendicular bisector ofAB¯¯¯¯¯¯¯¯ . List all the congruent segments.
 Is
C onAB¯¯¯¯¯¯¯¯ ? Why or why not?  Is
D onAB¯¯¯¯¯¯¯¯ ? Why or why not?
For Question 2, determine if
For Questions 37, consider line segment
 Find the slope of
AB .  Find the midpoint of
AB .  Find the equation of the perpendicular bisector of
AB .  Find
AB . Simplify the radical, if needed.  Plot
A,B , and the perpendicular bisector. Label itm . How could you find a pointC onm , such thatC would be the third vertex of equilateral triangle△ABC ? You do not have to find the coordinates, just describe how you would do it.
For Questions 812, consider
 Find the midpoint and slope of
AB¯¯¯¯¯¯¯¯ and use them to draw the perpendicular bisector ofAB¯¯¯¯¯¯¯¯ . You do not need to write the equation.  Find the midpoint and slope of
BC¯¯¯¯¯¯¯¯ and use them to draw the perpendicular bisector ofBC¯¯¯¯¯¯¯¯ . You do not need to write the equation.  Find the midpoint and slope of
AC¯¯¯¯¯¯¯¯ and use them to draw the perpendicular bisector ofAC¯¯¯¯¯¯¯¯ . You do not need to write the equation.  Are the three lines concurrent? What are the coordinates of their point of intersection (what is the circumcenter of the triangle)?
 Use your compass to draw the circumscribed circle about the triangle with your point found in question 11 as the center of your circle.
 Fill in the blanks: There is exactly _________ circle which contains any __________ points.
 Fill in the blanks of the proof of the Perpendicular Bisector Theorem.
Given:
Prove:
Statement  Reason 

1.  
2. 

3.  Definition of a midpoint 
4. 

5. 

6.  Reflexive PoC 
7. 

8. 
 Write a two column proof.
Given:
Prove:
Review (Answers)
To view the Review answers, open this PDF file and look for section 5.2.