Perpendicular Bisectors of a Triangle

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.

Perpendicular Bisectors of a Triangle Activity Grading Key

Point Values for Grading Problems

p. 3

Question 1

2 points for drawing
2 points for description

Question 2

1 point for prediction

p. 4

Question 3

2 points for drawing
2 points for description

Question 4

1 point for prediction

Question 5

2 points for drawing
2 points for description

p. 5

Question 6

3 points, (1 point for each description of acute, right and obtuse triangles)

p. 6

Question 7

2 points for drawing
2 points for description

p. 7

Question 8

2 points for drawing
3 points for description, (1 pt for circumscribed, 1 pt for center of circle, 1 pt for radius of circle)

Question 9

2 points for drawing of acute triangle
2 points for drawing of obtuse triangle
2 points for the description

p. 8

Question 10

1 point

Question 11

4 points, (1 point for each of the blanks)

Question 12

13 points, (2 points for definitions and Theorem, and 3 pts for properties of circumcenter in the acute, obtuse and right triangles)

p. 9

Problems

8 points, ( 1 point each segment length)

p. 10

Extension 1

4 points (2 points for drawing, 2 points for answer)

p. 11

Extension 2

6 points ( 2 points for drawing, 2 points for question a, 2 points for question b)

Total Points = 68 points

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.

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?

/ All three bisectors meet at the same point.
2 points

2 points

Question 2: What do you think will happen if the shape of the triangle is changed to an obtuse triangle?

Answers may vary (The perpendicular bisectors will still meet at the same point.)

1 point

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?

/ All three bisectors still meet at the same point, but in the acute triangle they meet at a point INSIDE the triangle; with the obtuse triangle they meet at a point OUTSIDE of it.
2 points

2 points

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?

Answers may vary (They will meet on the hypotenuse of the triangle.)

1 point

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?

/ All three bisectors still meet at the same point, but in a right triangle they meet at a point on the hypotenuse.
2 points

2 points

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.

The point of concurrency of the perpendicular bisectors, is the point where the three perpendicular bisectors of the triangle will intersect.

If the triangle is acute they will meet at a point inside the triangle.

If the triangle is obtuse they will meet at a point outside of the triangle.

If the triangle is a right triangle they will meet at a point on the hypotenuse.

3 points

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

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?

/ The distance from the point of concurrency to each of the vertices is the same.
2 points

2 points

Lets 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.

/ The triangle is circumscribed by the circle. The point of concurrency is the center of the circle, and the distance between the point of concurrency and the vertices is the radius of the circle.
3 points

2 points

Question 9: Grab onto one of the vertices and move it to change the shape of the triangle. Make twoadditional screen capturesto 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.

2 points 2 points
In each type of triangle, the distance between the point of concurrency and the vertices of the triangle will be the same. This distance is the radius of the circle, making the point of concurrency the center of the circle.
2 points

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.

Because it is the center of the circle that is circumscribed about the triangle.

1 point

Question 11: Finish the following theorem:

Circumcenter Theorem—Thethree _perpendicular bisectors__ of a triangle meet at the __circumcenter______, which is a point of concurrency that is __equidistant______to the __vertices______of the triangle.

4 points

Question 12: Write all of the definitions, properties and theorems you have learned in this exercise.

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. 2 points

Circumscribe: To draw a circle around a polygon such that all vertices of the polygon lie on the circle. 2 points

Point of Concurrency: three or more lines (or segments or rays) intersect at the point of concurrency. 2 points

Circumcenter: The circumcenter of a triangle is the point of concurrency of the perpendicular bisectors of the sides of a triangle. 2 points

Circumcenter Theorem: The three perpendicular bisectors of a triangle meet at the circumcenter, which is a point of concurrency that is equidistant to the vertices of the triangle. 2 points

The circumcenter is in the acute triangle; outside of the obtuse triangle; and on the hypotenuse of the right triangle. 3 points

Problems:

The perpendicular bisectors of ∆ABC meet at point D.

Find DB= 14

Find AE= 11

Find ED (Hint: Use the Pythagorean Theorem.) Write your answer in simplified radical form. = 5 √(3)

3 points /
R is the circumcenter of ∆OPQ. OS = 10,
QR = 12, and PQ = 22.

Find OP= 20

Find RP= 12

Find OR= 12

Find TP= 11

Find RT= √(23)

5 points /

Extension #1:

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.
You could connect the three points to form a triangle, and then construct perpendicular bisectors from each side of the triangle. They would meet at the point of concurrency, which is the circumcenter.
2 points
Make a sketch,by hand, of the situation. Indicate the best location for the new home.

2 points

Extension#2:

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 = 2.35702 ft = 28.28424 inches
2 points
b)Suppose the radius of the mycelium increases at a rate of about 8 inches per year. Estimate its age.
Age = 28.28424÷8 = 3.53553 years
2 points /

2 points

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