“Abridged” Lesson Plan & Goals
After this lesson, students should be able to:
 Find the circumcenter of a triangle
 Know the properties of the circumcenter of a triangle
 Circumscribe a circle about a triangle
Class Structure:


Student teams of 4 (grouped)
Each student gets a different triangle to construct perpendicular bisectors individually and then discuss
conjectures/questions as a group.
Teacher Role:
There are 4 distinct parts to the lesson:
 Construction of the perpendicular bisectors of the triangle
 Clarifying questions for students
 Debriefing questions
 Application problem
Construction of the perpendicular bisectors of a triangle (10 minutes)
Introduce the class to what they will be doing today (constructing perpendicular bisectors in a triangle, answering
questions, and finally culminating with a class project). Get a read on whether they will need a quick reminder on
how to do this and briefly go over the construction if necessary. A video segment is provided on the flipchart. (10
minutes)
 During construction of perpendicular bisectors,
o Facilitate construction of bisectors for groups
o Guides students towards the concept that it is just one intersection point, not multiple points
Clarifying questions for students (5-10minutes)
Bring class back together to briefly discuss findings and clarify discussion questions. Let students know that they
are to discuss the findings for each of their triangles as a GROUP in order to come up with generalizations about
triangle types (when necessary) as they go through the questions. (20 minutes)
 At Q&A time,
o Rotate around the room to guide and participate in discussions.
o Keep students on task by refocusing where they are supposed to be.
Debriefing questions (10 minutes)
Whether students have completed going through all the questions or not, start asking students to come to the
board to answer the questions. This will refocus students to completing the questions task since answers are
already being put on the board or discussed by you. DO NOT SPEND MORE THAN 10 MINUTES on this task. This
should be reviewing, supporting, and clarifying the conjectures they have already made.
Application Problem (20 minutes)
 For Question 3 (Presentation on Big Paper),
o Make sure students understand that the fish farm has to be “offshore” so that their island design
reflects this idea.
o Discuss with students the parts of the problem that are important so that they take these into
consideration for their presentation.
o Problem addresses triangle type and location of circumcenter as well as equal distances between
circumcenter and vertices of the triangle.
Timeline:
9:05 – 9:15
Construction
9:15 – 9:25
Clarifying Questions
9:25 – 9:45
Questions answered with Group
9:45 – 9:55
Question Debrief
9:55 – 10:15
Application Problem/Presentation Design on chart paper
10:15 – 10:35
Group Presentations
Circumcircle of a Triangle (10 minutes)
In triangle ABC, construct the perpendicular bisectors of segments AB, BC, and AC. Verify the following to make
sure your construction was done correctly:
 The angle created by the segment and the perpendicular bisector is 90°
 The lengths between the endpoints of the segment and the intersection point of the segment and the
perpendicular bisector are congruent.
Describe your findings and write a conjecture that reflects the findings for the group in reference to perpendicular
bisectors in any triangle at the bottom of the page. Then answer the questions on the answer sheet provided.
B
A
C
Circumcircle of a Triangle (10 minutes)
In triangle ABC, construct the perpendicular bisectors of segments AB, BC, and AC. Verify the following to make
sure your construction was done correctly:
 The angle created by the segment and the perpendicular bisector is 90°
 The lengths between the endpoints of the segment and the intersection point of the segment and the
perpendicular bisector are congruent.
Describe your findings and write a conjecture that reflects the findings for the group in reference to perpendicular
bisectors in any triangle at the bottom of the page. Then answer the questions on the answer sheet provided.
B
A
C
Circumcircle of a Triangle (10 minutes)
In triangle ABC, construct the perpendicular bisectors of segments AB, BC, and AC. Verify the following to make
sure your construction was done correctly:
 The angle created by the segment and the perpendicular bisector is 90°
 The lengths between the endpoints of the segment and the intersection point of the segment and the
perpendicular bisector are congruent.
Describe your findings and write a conjecture that reflects the findings for the group in reference to perpendicular
bisectors in any triangle at the bottom of the page. Then answer the questions on the answer sheet provided.
B
A
C
Circumcircle of a Triangle (10 minutes)
In triangle ABC, construct the perpendicular bisectors of segments AB, BC, and AC. Verify the following to make
sure your construction was done correctly:
 The angle created by the segment and the perpendicular bisector is 90°
 The lengths between the endpoints of the segment and the intersection point of the segment and the
perpendicular bisector are congruent.
Describe your findings and write a conjecture that reflects the findings for the group in reference to perpendicular
bisectors in any triangle at the bottom of the page. Then answer the questions on the answer sheet provided.
B
A
C
Look at the triangles of others in your group to see if they had similar findings as you answer the following
questions. Answers to your questions should reflect the answers for the GROUP after you have discussed your
individual findings.
Questions (20 minutes):
1.
Location of Intersection Point(s) of Perpendicular Bisectors in a Triangle
a. Where is/are the intersection point(s) for the perpendicular bisectors when the triangle is…
i. Acute:___________________________________________________
ii. Obtuse:__________________________________________________
iii. Right:____________________________________________________
b.
We say that these are points of concurrency. Create a definition for what a point of concurrency
is.
_____________________________________________________________________________________________
c.
Write a conjecture for each type of triangle that relates the type of triangle to the location of the
point of concurrency.
Example:
If the triangle is ____________, then point of concurrency lies____________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
_________________________________________________________________________________________.
2.
Distance of Concurrency Point from Vertices of a Triangle
a. Using your compass, compare the distances between the point of concurrency of the
perpendicular bisectors and the vertices of the triangle. Describe your findings and write a
conjecture based on your findings.
_____________________________________________________________________________________________
_____________________________________________________________________________________________
_____________________________________________________________________________________________
_____________________________________________________________________________________________
b.
Now, use a ruler to measure the distance to the nearest tenth of a centimeter. Was your
conjecture correct? Explain in detail and show your measurements on the construction.
_____________________________________________________________________________________________
_____________________________________________________________________________________________
_____________________________________________________________________________________________
_____________________________________________________________________________________________
Circumcircle of a Triangle
c. Would your triangle “fit” inside a circle with all its vertices on the circle?
_____________________________________________________________________________________________
d.
With your compass, use the point of concurrency of the perpendicular bisectors as a center and
one of the vertices as a radius to construct a circle about (or around) your triangle. Describe your
findings.
_____________________________________________________________________________________________
_____________________________________________________________________________________________
_____________________________________________________________________________________________
_____________________________________________________________________________________________
This is called the “circumscribed circle or circumcircle of the triangle.”
e. Would any triangle “fit” inside a circle with all its vertices on the circle? Support your conjecture
by providing an explanation of why you think you are correct.
_____________________________________________________________________________________________
_____________________________________________________________________________________________
_____________________________________________________________________________________________
_____________________________________________________________________________________________
_____________________________________________________________________________________________
f.
This point of concurrency, where the perpendicular bisectors meet, seems very important. If the
circle that surrounds the triangle is called the “circumscribed circle” and the point of
concurrency of the perpendicular bisectors is the “center” of that circle, then what should we
call the point of concurrency where the perpendicular bisectors meet?
_____________________________________________________________________________________________
3.
Application: Fish Farming Tribes of Kona Island (30
minutes)
Kona Island has 3 fishing tribes on it, the Cordos, the Bonte, and the Meshoa, all living in different parts of
the island. In order to maximize their efforts for sustaining their people, tribal elders have decided to use
state-of-the-art technology to start a fish farm offshore. Your team of engineers has been hired by the
tribes to locate an equitable location for the fish farm. Your task is to present to the tribal elders the
location of the farm and explain to them why it is an equitable location. But be careful…
If you are off the mark…
YOUR TEAM goes in the pot with the FISH STEW!
Make sure to include: a drawing of the island, the location of the villages for each tribe and BOTH a visual
representation of why this is an equitable offshore location, and a written explanation of how you would
find the point.
Reflections on the Lesson


Students MUST know how to construct a perpendicular bisector to a segment. Otherwise, students will
spend too much time on this part. Ensure students are aware of this construction with a little demo if
necessary, and let them know that they will be doing EXACTLY the same thing to all 3 segments of the
triangle.
Time is of the essence…you must keep to the timeline in order to have enough time for the application
problem and presentations.