Eligible Content M11.C.1.2.1 Identify and/or use properties of triangles.
This lesson is planned to provide a hands-on discovery approach to the points of concurrency in a triangle. Each station requires the use of safety compasses, rulers, scissors, construction paper and/or miras.
The room is divided into five stations, each station set up for the construction of one of the four constructions plus an additional one for students to work on cumulative review for the mid-term exam. The students are divided into mixed ability groups of 4 based on a pre-assessment of their abilities based on knowledge of the terms altitude, perpendicular bisector, and median.
Each group is assigned a “leader” who is responsible for communicating with the teacher when the group is not able to answer a question or construction. The time limit per station is given at the beginning of the class and indicated changing times are listed on the board. The teacher will remind the students when they have 3-5 minutes left at any station. If students are not able to complete the entire station packet in the time allotted, they will be able to complete the remaining questions on their own for homework.
______Name ______
Chapter 5 Lesson 3 Pre-Assessment
Name each figure in ∆BDF.
1. an angle bisector.
2. a median
3. a perpendicular bisector
4. an altitude
Station 1: Medians of a Triangle
Trace and cut out one of each triangle. You should have an acute (#1), obtuse(#2) and right(#3) triangle when you are finished. Number each triangle.
Construct the Centroid of each Triangle by Constructing the Three Medians
Find the midpoint of one side of the triangle by matching up two vertices and folding a crease at the midpoint. Using a ruler, draw a line connecting this midpoint to the opposite vertex. Repeat for each side.
Where these three line segments intersect is called the point of concurrency. Mark this point with a pen and label it point C.
The point of concurrency of the three medians of triangle is called the ______.
Properties of a Centroid
1. Using a ruler on the first triangle, measure the length of one of the medians.
2. Using a ruler on the same triangle and same median:
- Measure the length of the line segment from the vertex to the centroid.
- Measure the length of the line segment from the centroid to the midpoint.
- Do this for all three medians.
Column 1 / Column 2 / Column 3 / Column 4 / Column 5
Triangle Number / Measure of Median / Distance from vertex to Centroid / Distance from Centroid to Midpoint /
1
1
1
2
2
2
3
3
3
Practice Problems
In ∆TUV, Y is the centroid.
1. If YW = 9, find TY and TW.
2. If YU = 9, find ZY and ZU.
3. IF VX = 9, find VY and YX.
Name the point of concurrency of the angle bisectors.
4. 5.
6. Multiple Choice. C is the centroid of ∆DEF. If GF = 6x2 + 9y, which expression represents CF?
a) 2x2 + 9y b) 2x2 + 3y
c) 6x2 + 9y d) 4x2 + 6y
Station 2: Altitudes of a Triangle
Construct the Orthocenter of a Triangle by Constructing the Three Altitudes
1. Position the mira so that it is perpendicular to one side of the triangle and such that it also goes through the opposite vertex. (you may need to extend the side of the triangle)
2. Construct a line segment.
You have just constructed an Altitude of the triangle.
3. Repeat this for the other two sides and vertices.
Where these three line segments intersect is called the point of concurrency.
The point of concurrency is a point where ______or more lines intersect.
The point of concurrency of the three altitudes of triangle is called the ______.
4. Construct the three altitudes of the other two triangles. (**Are you running into any problems with some of the altitudes? If the perpendicular segment is away from the triangle, use a pencil to extend the side of the triangle.)
Properties of an altitude.
1. If you travel a path that is perpendicular to a certain location, you have traveled the ______(minimum or maximum) distance.
2. The altitude of a triangle can also be referred to as the ______of the triangle.
3. When have you used the altitude of a triangle?
Station 3: Angle Bisectors of a Triangle
(You will need construction paper, protractor, ruler, compass and scissors)
Trace and cut out one of each triangle. You should have an acute (#1), obtuse(#2) and right(#3) triangle when you are finished. Number each triangle.
Construct the Incenter of each Triangle by Constructing the Three Angle Bisectors
Use paper folding to find the three ANGLE BISECTORS of each triangle by matching up the two sides of the triangle and folding down the middle. Repeat for each side.
Where these three line segments intersect is called the point of concurrency. Mark this point with a pen and label it point I.
The point of concurrency of the three Angle Bisectors of triangle is called the ______.
Properties of the Incenter
1. Using a protractor, construct a segment from the Incenter that is perpendicular to each side of the triangle.
2. Using a ruler, measure each of these segments that you just created.
3. What do you notice?
4. What can you conclude about your observation?
5. Place the center of the safety compass on the incenter. Try to construct a circle that is inside of the triangle such that the circle touches all three sides and does not go outside of the triangle.
Can it be done?
The circle is ______in the triangle.
Repeat this for all three triangles.
Practice Problems
APPLYING THOEREM 5-7
Station 4: Perpendicular Bisectors of a Triangle
Construct the Circumcenter of a Triangle by Constructing Perpendicular Bisectors
1. Position the mira so that it is perpendicular to one side of the triangle and such that it also goes through the midpoint of that side. (the side will reflect upon itself and the endpoints will match up)
2. Construct a line segment.
You have just constructed one of three perpendicular bisectors of the triangle.
3. Repeat this for the other two sides and vertices.
Where these three line segments intersect is called the point of concurrency.
The point of concurrency is a point where ______or more lines intersect.
The point of concurrency of the three perpendicular bisectors is called the ______.
4. Construct the 3 perpendicular bisectors in each of the two remaining triangles.
Properties of the Circumcenter
1. Using the safety compass, construct a circle that includes all three vertices of the triangle. What do you notice about the center of the circle?
Where do you need to put the center of your safety compass?
We say the circle you just created is ______about the triangle.
1. Construct line segments from the circumcenter to each vertex.
2. Using a ruler, measure each new segment.
What do you notice about the length of each of these new segments?
Practice Problems
Exit Ticket Name: ______
Date: ______Period: _____
In your own words, how would you describe the following to your relatives over Thanksgiving dinner? Please use pictures to help with your description.
1. Concurrent Lines
2. Median
3. Altitude
4. Perpendicular Bisector of a Segment
5. Angle Bisector
6. Centroid
7. Circumcenter
8. Incenter
9. Orthocenter
10. Circumscribe
11. Inscribe
12. Tell me a fact about centroids and a fact about medians of a triangle.
13. Tell me something that you consider unique about altitudes of a triangle.
14. Tell me what you found about the point of concurrency of all three perpendicular bisectors of a triangle.
15. Tell me what you found about the point of concurrency of all three angle bisectors of a triangle.