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Kuzas, Richmond, Tocchi – Team 6 – Circumcenter
Lesson 1: The perpendicular bisector to a segment
Strand:
Learning Objectives:
1. To discover the definition of perpendicular bisector to a segment.
2. To analyze what property characterizes the points on the perpendicular bisector, as well as
those points in the half planes that the perpendicular bisector defines.
Materials: Computer lab or a set of calculators equipped with Cabri Geometry II. A sheet of
paper.
Procedure: (suggestions)
1. Group students using the method of choice.
2. Pose the questions:
a. Fold the sheet of paper so that opposite corners and top and bottom edge coincide. Open
the paper and try to discover how does the line defined by the crease relate to the edges
that it touches.
The line forms a 45 degree angle with the top and bottom sides of the paper.
b. Draw a point on the line defined by the crease and measure the distance from this point to
the corners of the paper. Draw some other points on the same line and repeat the same
measurements. What do you notice about these measurements?
The measurements add up to the same number.
Assessment: Is left up to the instructor. But keep in mind that authentic forms of assessment are
usually best when conducting a lesson that relies heavily on discovery based learning.
Kuzas, Richmond, Tocchi – Team 6 – Circumcenter
Lab 1. Perpendicular bisector
Goal: Construct the perpendicular bisector of a segment and analyze its properties.
Investigate using Cabri:
1. Construct segment AB .
[Use segment tool]
2. Construct the perpendicular bisector of segment AB and call this line l.
[Use perpendicular bisector tool]
3. Find the intersection point of AB and l and label it C.
[Use intersection points tool]
Let us analyze the properties of l:
4. Measure CA and CB . What can you say about the values of CA and CB found?
Which point of AB is C?
[Use distance & length tool]
The values of CA and CB are equal, so, C is the midpoint of AB.
5. Create point P on l. Measure PCA and PCB . What do you observe? How are l and
AB related?
[Use angle tool]
PCA and angle PCB are both 90o. This means that l and AB are perpendicular to one
another. Since they intersect, they are perpendicular bisectors.
Define the perpendicular bisector of a segment using your observations in 4 & 5.
Perpendicular Bisector- The perpendicular bisector of a segment is the intersection of two lines
that form 90 degree angles at their intersection.
6. Measure segments PA and PB and write their values PA and PB. How are these two
measurements related? Grab point P and move it along l, what do you observe that
happens to PA and PB?
PA =7.56 units, PB =7.56 units. As I move P along l, the values of PA and PB change
simultaneously. As I move P closer to C, the values get smaller, and vice versa. PA and PB
always remain equal.
Kuzas, Richmond, Tocchi – Team 6 – Circumcenter
7. Create point Q above AB to the left of l and point R above AB to the right of l.
8. Measure segments QA and QB and write their values QA and QB. How are these two
measurements related? Grab point Q and move it in the region to the left of l, what do
you observe that happens to QA and QB?
QA = 2.68 units, QB = 8.80 units; these measurements are not as related as our previous
measurements, which were constantly equal. This is because I randomly placed Q to the left of 1.
As I move Q in the region to the left of l, QA always remains less than QB. As I move Q closer
to A, the distance QA becomes smaller (obvious). However, as I move Q to the right of l, QA
increases and QB decreases.
9. Measure segments RA and RB and write their values RA and RB. How are these two
measurements related? Grab point R and move it in the region to the right of l, what do
you observe that happens to RA and RB?
RA = 7.22 units, RB = 4.56 units; these two measurements are related in the same way as
described in #8. In this instance, the length of QA is always greater than QB.
Lab 1: Perpendicular Bisector
Summarize in your own words the properties that you have discovered.
We have discovered that two lines or line segments are perpendicular bisectors of one another
when two perpendicular lines intersect. Without using the word perpendicular in the definition of
perpendicular bisector however, we can say that it is when two lines or line segments intersect
and form four 90 degree angles at their intersection. If a point P is on a line AB and the distance
between P and A is equal to the distance between P and B, the P is the midpoint of AB.
Kuzas, Richmond, Tocchi – Team 6 – Circumcenter
Lab 2. The Circumcenter and the Circumcircle of a triangle
Definition. We call the circle that passes through the three vertices of a triangle its
circumcircle, and we refer to the center of this circle as the circumcenter of the triangle.
Goal. To find the Circumcenter and the Circumcircle of a triangle.
Using Cabri:
1. Draw any triangle ABC. Draw the perpendicular bisector to the sides AB and BC ,
and call O the point where they meet.
2.
Since O is on the perpendicular bisector to AB , what must be true about OA and
OB? Measure OA and OB and confirm your hypothesis.
The distance between O and A must be equal to the distance between O and B. This is
true.
3. Similarly, since O is on the perpendicular bisector to BC , what must be true about
OB and OC? Measure OB and OC and confirm your hypothesis.
These distances must also be equal. They are.
4. What can you conclude about the distances OA, OB, and OC?
We can conclude that all three distances, OA, OB, and OC, are all equal.
5. If you draw a circle with center O and radius OA, will the circle pass through B and
C? Why?
Yes, the circle will pass through B and C because OA is the length of the radius. OA is
equal to OB and OC, so the points B and C will also be on the circle
6. Should point O be on the perpendicular bisector to the side AC ? Yes O should be
on the bisector because we already know that the distance between OC = OA. The
distance between O and each of the points A, B, and C is like a radius of a circle.
Draw the perpendicular bisector to side AC and confirm your conclusion.
Kuzas, Richmond, Tocchi – Team 6 – Circumcenter
Lab 2: Circumcenter and Circumcircle
In your own words: How do we find the circumcenter and the circumcircle of a given triangle?
To find the circumcenter, perpendicularly bisect two of the sides of the triangle. Where the two
lines intersect is the circumcenter. The circumcircle is formed by drawing a radius from the
circumcenter to any vertex, then use a compass to draw the circle.
Extension. The mayors of three cities are pulling resources together to dig a well that will
provide drinking water for the three cities. Since each city is contributing with the same amount
of money, where should the well be situated so that its distance to each city is the same?
The circumcenter!
Journal Activity
Circumcenter & Circumcircle of a Triangle
1. List all definitions and properties that you have learned in this activity.
perpendicular bisector – The perpendicular bisector of a segment is a line that splits the
segment into two equal parts and forms 90 degree angles at their intersections.
circumcenter – The point created when perpendicular bisectors of two triangle sides
intersect.
circumcircle – The circle created around the circumcenter and includes the vertices of the
triangle.
2. Can you think of any applications of this topic?
Yes, we saw an example in the extension of how this topic could be used. We could also
use this idea to find the epicenter of an earthquake.
3. Can you relate this topic/concepts with other(s) previously studied? Explain your answer.
No, we have not had other concepts to compare it to so far.