What do you mean . . .

A Triangle Has More Than One Center?!?

A Dynamic Geometry Lesson for High School Students

Designed by
Sarah Grabowski

Last updated November 26, 2002.


Instructional Goals

After completion of this lesson students will be able to:

·  Define the four centers of a triangle

·  Describe how to construct each of the four centers of a triangle

·  Summarize the theorems associated with the four centers of a triangle

·  Explain a use for each of the four centers outside the mathematics classroom

Performance Objectives

·  Students will be able to list and define the four centers of a triangle without assistance with 100% accuracy.

·  Given three points, students will be able to construct the centroid of that triangle with 100% accuracy.

·  Given three points, students will be able to construct the circumcenter of that triangle with 100% accuracy.

·  Given three points, students will be able to construct the incenter of that triangle with 100% accuracy.

·  Given three points, students will be able to construct the orthocenter of that triangle with 100% accuracy.

·  Given the title Triangle Centroid Theorem, students will be able to summarize the theorem with 90% accuracy.

·  Given the title Triangle Circumcenter Theorem, students will be able to summarize the theorem with 90% accuracy.

·  Given the title Triangle Incenter Theorem, students will be able to summarize the theorem with 90% accuracy.

·  Given the title Triangle Orthocenter Theorem, students will be able to summarize the theorem with 90% accuracy.

·  Given real life situations, students will be able to decide which center is most appropriate and then solve the problems with 85% accuracy.


Learner Prerequisites

Prerequisite Math Skills

·  Students are able to define the geometric words:

o  triangle

o  circle

o  segment

o  side

o  angle

o  vertex (vertices)

o  midpoint

o  median

o  perpendicular bisector

o  angle bisector

o  altitude

o  concurrent

o  theorem

·  Students are able to construct the ______of a triangle.

o  medians

o  perpendicular bisectors

o  angle bisectors

o  altitudes

Prerequisite Computer Skills

·  Students are able to demonstrate basic computer skills (i.e. opening a file, saving and printing their work, using the keyboard and the mouse, etc.).

·  Students are able to demonstrate basic use of the program Geometer's Sketchpad (i.e. constructing points and segments, labeling objects, etc.).


Materials & Resources

Classroom Materials

·  a variety of different sized and angled triangles cut out from sturdy cardboard (enough for every student to have one)

·  one medium sized circle cut out from sturdy cardboard

·  a box of brand new, never been used, regular, old number two pencils (enough for every student to have one)

·  a box of colored pencils (enough for each student to use four different colors, but sharing will work fine)

·  a compass (one for each student)

·  a ruler (one for each student)

·  scissors (one for each student)

·  thumbtacks (enough for each student to have four)

·  string (enough for each student to have about three feet)

·  a blackboard or markerboard

·  chalk or dry-erase markers

·  vocabulary review worksheet (download below)

·  end of lesson assessment (download below)

Computer Resources

·  computer lab (with enough computers for half of the class)

·  teacher computer hooked up to an overhead or projector

·  computer disks or a server for students to save their work

·  Geometer's Sketchpad software (on all computers)

·  gsp files and scripts (download below)

·  Adobe Acrobat Reader Software

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

A teacher's demo version of Geometer's Sketchpad can be downloaded from
Key Curriculum Press at http://www.keypress.com/sketchpad/sketchdemo.html.

Adobe Acrobat Reader can be downloaded from Adobe.com at
http://www.adobe.com/products/acrobat/readstep.html.

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *


Downloadable Resources

·  Entire Lesson Plan in Word format

·  Vocabulary Review Sheet

·  End of Lesson Assessment

·  GSP files & scripts

o  centroid file

o  centroid script

o  circumcenter file

o  circumcenter script

o  incenter file

o  incenter script

o  orthocenter file

o  orthocenter script

o  all four centers in one triangle file

o  all four centers in four different triangles file


Lesson Strategies

Introduction

  1. Divide the class into groups with 2 - 3 members in each group.
  2. Hand out the vocabulary review sheet, one to each student.
  3. Allow about 10 minutes for the groups to fill in the worksheet without any assistance.
  4. Then allow another 10 minutes for groups to use their notes/textbooks to check their answers (but not other groups).
  5. Walk around and offer assistance if needed.

Activity 1 - The Centroid

  1. Gain the students attention by holding up a cicrle cut from sturdy cardboard.
  2. Ask the students where the center of gravity is for the circle (answer: the center).
  3. Next hold up a triangle cut from sturdy cardboard.
  4. Ask the students where the center of gravity is for a triangle.
  5. After students have given some conjectures, tell them it is time to investigate where the center of gravity for a triangle is.
  6. Pass out cardboard triangles, new pencils, compass, ruler, scissors, 3 feet of string, 4 thumbtacks and 4 colored pencils to every student.
  7. The students should use the new pencils to balance the triangle and locate the center of gravity.
  8. Then use a colored pencil to mark the center. Make sure students keep track which center is which color.
  9. Is there only one center?
  10. Using the supplies passed out, students now try to locate the exact center of gravity by using mathematics.
  11. Offer a hint: Use the words on the vocabulary review sheet for ideas.
  12. After students make conjectures, ask them if their conjectures work for all of the triangles in their group.
  13. Next ask if their conjecture would work for all triangles.
  14. In order to test their conjectures further, students now will work with GSP.
  15. Everyone should start with a triangle in GSP.
  16. Have students test their conjectures. But how do we check for the center of gravity in a 2-dimensional setting?
  17. Lead students into figuring out that the center of gravity is such because there is equal area around that center.
  18. Students hopefully have figured out by now that the center is found by constructing the medians of the traingle and locating their intersection.
  19. Using two of the vertices and the center, a new traingle is formed. By using each combination of two vertices and the center, 3 traingles are formed (i.e., ABcenter, BCcenter, CAcenter).
  20. The areas in each of these traingles will be equal if the center found is the true center of gravity.
  21. To check, construct the Polygon Interior located under the Construct Menu.
  22. Then select the new area and measure the Area under the Measure Menu. Do this for each of the 3 triangles. The areas should all be equal!
  23. Tell students to move the vertices of their triangle to different places, thus creating different triangles.
  24. Does their conjecture still hold? Are the areas still equal?
  25. Bring the class back together and form Theorem 1:

Every triangle has three medians. In any triangle, the three medians intersect at a point, and only one point, which is always inside the triangle. The point of intersection of all three medians is called the centroid of the triangle. The centroid is also the center of gravity.


Activity 2 - The Circumcenter

  1. Gain the students attention by holding up a triangle cut from sturdy cardboard.
  2. Ask the students to find the point that is exactly the same distance from each of the vertices to that point.
  3. After students have given some conjectures, tell them to investigate where this new center is.
  4. Using the supplies passed out, students now try to locate the exact center by using mathematics.
  5. Offer a hint: Use the words on the vocabulary review sheet for ideas.
  6. Then use a different colored pencil to mark the center. Make sure students keep track which center is which color.
  7. Is there only one center?
  8. After students make conjectures, ask them if their conjectures work for all of the triangles in their group.
  9. Next ask if their conjecture would work for all triangles.
  10. In order to test their conjectures further, students now will work with GSP again.
  11. Everyone should start with a triangle in GSP.
  12. Have students test their conjectures. But how do we check for the center in a 2-dimensional setting?
  13. Students hopefully have figured out by now that the center is found by constructing the perpendicular bisectors of the traingle and locating their intersection.
  14. Pick any vertex and then the center and construct a Segment under the Construct Menu.
  15. Do this for the other two vertices.
  16. Then select one of the new segments and measure the Length under the Measure Menu. Do this for each of the 3 new segments. The Lenghts should all be equal!
  17. Tell students to move the vertices of their triangle to different places, thus creating different triangles.
  18. Does their conjecture still hold? Are the lengths still equal?
  19. Tell students to select the center and then one of the new segments.
  20. Under the Construct Menu, select construct a Circle by Center and Point.
  21. Ask the students if they noticed anything. The circle goes through each of the vertices.
  22. Then ask the students why?
  23. Next form Theorem 2:

The circumcenter of a triangle is the point in the plane equidistant from the three vertices of the triangle. The three perpendicular bisectors of a triangle intersect at one point to construct the circumcenter of a triangle. The circumcenter of a triangle does not necessarily exist in the interior of the triangle. Often the perpendicular bisectors of a triangle intersect outside the triangle. The circumcenter is the center of the circumcircle of the triangle. The circumcircle goes through all three vertices of the triangle.


Activity 3 - The Incenter

  1. Gain the students attention by holding up a triangle cut from sturdy cardboard.
  2. Ask the students to find the point that is the center of the largest possible circle that will completely fit inside the triangle..
  3. After students have given some conjectures, tell them to investigate where this new center is.
  4. Using the supplies passed out, students now try to locate the exact center by using mathematics.
  5. Offer a hint: Use the words on the vocabulary review sheet for ideas.
  6. Then use a different colored pencil to mark the center. Make sure students keep track which center is which color.
  7. Is there only one center?
  8. After students make conjectures, ask them if their conjectures work for all of the triangles in their group.
  9. Next ask if their conjecture would work for all triangles.
  10. In order to test their conjectures further, students now will work with GSP again.
  11. Everyone should start with a triangle in GSP.
  12. Have students test their conjectures. But how do we check for the center in a 2-dimensional setting?
  13. Students hopefully have figured out by now that the center is found by constructing the angle bisectors of the traingle vertices and locating their intersection.
  14. Pick any vertex and construct the Angle Bisector under the Construct Menu.
  15. Do this for the other two vertices. And find the intersection point.
  16. Drop perpendicular lines from the center to each of the traingle sides to create new segments.
  17. Then select one of the new segments and measure the Length under the Measure Menu. Do this for each of the 3 new segments. The Lenghts should all be equal!
  18. Tell students to move the vertices of their triangle to different places, thus creating different triangles.
  19. Does their conjecture still hold? Are the lengths still equal?
  20. Tell students to select the center and then one of the new segments.
  21. Under the Construct Menu, select construct a Circle by Center and Point.
  22. Ask the students if they noticed anything. The circle goes through each of the dropped perpendiculars and hit the sides in exactly one place.
  23. Then ask the students why?
  24. Next form Theorem 3:

The incenter of a triangle is the point on the interior of the triangle that is equidistant from the three sides. The angle bisectors of a triangle intersect each other at a point to construct the incenter of the triangle, which is always in the interior of a triangle. The incenter is the center of the incircle.


Activity 4 - The Orthocenter

  1. Ask students to look at their review sheet and see what the remaining constructiond for triangle centers is remaining (answer: altitude).
  2. Have students construct the altitude for each side of a triangle constructed in GSP .
  3. Do the altitudes intersect? Not necessarily.
  4. Extend the altitudes into lines. Do the extended altitudes intersect?
  5. Is there only one intersection point?
  6. Tell students to move the vertices of their triangle to different places, thus creating different triangles.
  7. Does their conjecture still hold?
  8. Bring the class back together and form Theorem 4:

The lines containing the altitudes of a triangle meet at one point called the orthocenter of the triangle. Because the orthocenter lies on the lines containing all three altitudes of a triangle, the segments joining the orthocenter to each side are perpendicular to the side. Altitudes are segments and are not necessarily concurrent; the lines that contain the altitudes are the only guarantee. Thus the orthocenter isn't necessarily in the interior of the triangle.

Conclusion

  1. Hand out the end of lesson assessment, one to every student.
  2. If time permits allow students to work together still in their groups, otherwise the assignment is take-home.
  3. Grade the assessment according to the rubric, which is out of 25 points.

Possible Extensions

·  Formal Proofs of the 4 Theorems

·  Concurrency Proofs

·  What happens to the centers specifically when the triangle is right, acute, or obtuse?