Project AMP
Dr. Antonio R. Quesada Director, Project AMP
Perpendicular Bisectors of a Triangle Activity—Teacher Notes
Objectives:
 Students will locate circumcenter of a triangle
 Students will discover that the circumcenter is equidistant from the vertices
of the triangle
 Students will find measures of segments given a triangle with the
perpendicular bisectors drawn in
 Students will discover the location of the circumcenter based on the
classification of the triangle by angle
Standard and Indicator:
Geometry and Spatial Sense, #1: Formally define and explain key aspects of
geometric figures: including
a) Interior and exterior angles of polygons
b) Segments related to triangles
c) Points of concurrency related to triangles
d) Circles
Prior Knowledge:
 Students must know the classifications of a triangle by angle; acute, obtuse,
and right.
 Students must be familiar with the terms: perpendicular, bisector and
midpoint
 Students need to know and apply the Pythagorean Theorem.
 Students need to know how to simplify radicals.
Materials Needed:
 TI-Nspire calculator
 Pencil
 Straight-edge
 Worksheet
Students must have a basic knowledge of the NSpire calculator. Most actions are
explained in the lesson, but students need to know how to open graph pages, and
how to manipulate points with the “grab” feature.
This activity could easily be adapted for use in other Geometry software, such as
GeoGebra or Geometers Sketchpad.
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Project AMP
Dr. Antonio R. Quesada Director, Project AMP
Perpendicular Bisectors of a Triangle
In this activity we will explore what happens when we construct the three
perpendicular bisectors of the sides of an acute triangle, a right triangle, and an
obtuse triangle.
Before you begin, you must make sure your calculator is in degree mode and not
in radian mode. Do this by pressing the home key and selecting 8: System Info,
then 2: System Settings to verify that you are in the correct mode.
Step one: Construct a Triangle
a. First open a page in Graphs and Geometry on your calculator. Do this by
pressing the home key, then selecting 2:Graphs and Geometry.
b. Next, press the menu key and select 8:Shapes, then choose 2:Triangle.
c. Now you will be back on your graph page with the pencil icon. Drop three
points at different places on the graph. Do this by moving the pencil and
pressing the select key.
Step two: Find the Measures of the Angles of the Triangle.
a. Press menu again, and select 7:Measurement, then 4:Angle.
b. Back in the graph screen, move the cursor until it is over a point, then select
the point. Move the cursor to another point and select that point, (this point will
be the vertex of the angle you are measuring) and finally to the third point and
select it, then press select again. You will now see the measure of the angle on
the screen. Escape out of measuring mode. Grab and drag the angle measure
next to the angle it is measuring.
c. Repeat this process until you have measured all three angles.
Now we are ready to begin the lesson. We will start with an acute triangle. Look
at your screen to determine if you already have an acute triangle. Remember
that you have the measures of all three angles to examine.
If you do not have an acute triangle, you will change it into an acute triangle by
grabbing onto one of the vertices and moving it. As you move it, you will notice
that the angle measures change as you change the shape of the triangle.
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Project AMP
Dr. Antonio R. Quesada Director, Project AMP
Definition: Perpendicular Bisector—The perpendicular bisector of a triangle is
a line (or ray or segment) that is perpendicular to the side of a triangle at the
midpoint of the side.
Definition: Circumscribe—To draw a circle around a polygon such that all
vertices of the polygon lie on the circle. (Hint: You will need to use this word
later in this activity.)
Now we will construct the perpendicular bisectors of each of the three sides of
the triangle. Here is how we will do it: First press the menu key and select
9:Constructions, then 3:Perpendicular Bisector. Back in the graph window,
move the pencil icon close to one of the sides of the triangle and you will see the
perpendicular bisector appear on that side. Press select when you see it
appear. Repeat the process until you can see the perpendicular bisector of all
three sides of the triangle. Press escape to get out of construction mode.
You may want to hide the axes of the graph so that you will be able to better see
the triangle and the segments. You can do this by pointing to the axes, pressing
the menu key, selecting 1:Actions, then 2:Hide/Show. Press select when you
have highlighted the axes, and they will be hidden, then escape from this mode.
Question 1: Insert a screen capture from your calculator. What do you notice
about the three perpendicular bisectors that you have constructed in your acute
triangle?
Question 2: What do you think will happen if the shape of the triangle is changed
to an obtuse triangle?
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Project AMP
Dr. Antonio R. Quesada Director, Project AMP
Now we will see what happens if we make our triangle an obtuse triangle. Move
the pointer to one of the vertices, and grab it. Move the point while watching the
measures of the angles. Stop when you have created an obtuse triangle. Notice
that the perpendicular bisectors you have constructed will move also, but they
will continue to be the perpendicular bisectors of the sides.
Question 3: Insert a screen capture from your calculator. What do you notice
about the three perpendicular bisectors that you have drawn in the triangle?
What is different from what you saw in the acute triangle?
Question 4: What do you think will happen if the shape of the triangle is
changed to a right triangle? Where do you think that the perpendicular bisectors
might meet?
Next we will see what happens if we make our triangle a right triangle. Again
grab one of the vertices of the triangle and move it until one of the angles
measures 90 degrees (or as close to 90 degrees as you can make it).
Question 5: Insert a screen capture from your calculator. What do you notice
about the three perpendicular bisectors that you have drawn in the triangle?
What is different from what you saw in the acute and the obtuse triangles?
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Project AMP
Dr. Antonio R. Quesada Director, Project AMP
Definition: Concurrent Lines—three or more lines (or segments or rays) that
intersect in the same point. That point is called the point of concurrency.
Question 6: Generalize what you know about the point of concurrency of the
perpendicular bisectors of an acute triangle, a right triangle, and an obtuse
triangle.
Now we will explore a characteristic of the point of concurrency of the angle
bisectors of a triangle. Let’s go back to our triangle in the calculator. We will
measure the distance between the point of concurrency and the vertices of the
triangle. First we must put a point at the point of concurrency.
Press the menu button on your calculator and choose 6:Points and Lines, then
3:Intersection Points. Back in your graph screen, point and select each of the
perpendicular lines. This will place a point at the point of concurrency. Press
escape to leave this mode.
Next we will measure the distance between the point of concurrency and each of
the three vertices. Here is how we will do it. First, press the menu button and
select 7:Measurement, then 1:Length. Back in the graph screen point at the
point of concurrency and select it, then move the cursor to one of the vertices
and select it. Hit the select button again and you can see the measure of that
distance. Repeat the instructions to find the distance from the point of
concurrency to each of the other vertices. Then escape from the measurement
mode.
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Project AMP
Dr. Antonio R. Quesada Director, Project AMP
Question 7: Insert a screen capture of the calculator. What can you note about
the distances from the point of concurrency of the perpendicular bisectors and
the vertices of the triangle?
Let’s explore further. Press menu again and select 8:Shapes, then 1:Circle.
Back in the graph screen point at the point of concurrency and select it. Then
move the cursor to one of the vertices and select it, then escape.
Question 8: Insert a screen capture of the calculator. Describe what you see.
Question 9: Grab onto one of the vertices and move it to change the shape of
the triangle. Make two additional screen captures to show all three triangles,
acute, obtuse and right (when included with your screen capture from Question
6). Describe what happens to the distance between the point of concurrency and
each of the vertices, and to the circle in relationship to the triangle.
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Project AMP
Dr. Antonio R. Quesada Director, Project AMP
Definition: Circumcenter of a Triangle—The circumcenter of a triangle is the
point of concurrency of the perpendicular bisectors of the sides of a triangle.
Question 10: Why do you think this point of concurrency is called the
circumcenter? Look at your drawings from Questions 6 and 7.
Question 11: Finish the following theorem:
Circumcenter Theorem—The three __________________ of a triangle meet at
the ______________________________, which is a point of concurrency that is
__________________________ to the _____________________ of the triangle.
Question 12: Write all of the definitions, properties and theorems you have
learned in this exercise.
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Project AMP
Dr. Antonio R. Quesada Director, Project AMP
Problems:
The perpendicular bisectors of ∆ABC meet at
point D.
a. Find DB
b. Find AE
c. Find ED (Hint: Use the Pythagorean
Theorem.) Write your answer in simplified
radical form.
R is the circumcenter of ∆OPQ. OS = 10,
QR = 12, and PQ = 22.
a. Find OP
b. Find RP
c. Find OR
d. Find TP
e. Find RT
Extension #1:
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Project AMP
Dr. Antonio R. Quesada Director, Project AMP
Your family is considering moving to
a new home. The diagram shows
the locations of where your parents
work and where you go to school.
The locations form a triangle.
In this diagram, how could you find a point that is equidistant from each
location? Explain your answer.
Make a sketch, by hand, of the situation. Indicate the best location for the
new home.
Extension#2:
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Project AMP
Dr. Antonio R. Quesada Director, Project AMP
A mycelium fungus grows underground in all directions from a central point.
Under certain conditions, mushrooms sprout in a ring at the edge. The radius
of the mushroom ring is an indication of the mycelium’s age.
Suppose three mushrooms in a
mushroom ring are located as shown.
Make a table in your calculator and enter
the points in the diagram. Make a
scatterplot of the points A, B, and C.
Connect the points to make triangle ABC.
Each unit will represent 1 foot.
Use the NSpire to graph and solve the
problem. Include a screenshot of the
calculator with your answers.
a) Find the radius of the mushroom
ring. (Hint: find the circumcenter.)
Radius = ________________
b) Suppose the radius of the
mycelium increases at a rate of
about 8 inches per year. Estimate
its age.
Age = _______________
References, Larson, Boswell, Stiff, Geometry, McDougal Littell, ©2004
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