Final Exam, MTH 606 (SU 09)
75 Points Possible
Name: _____________________________
Directions
Test Sections. This exam is organized in three sections: (1) Class Modules, a section covering teaching modules
from the first three weeks of class; (2) Project Modules, a section covering modules presented by classroom
teachers in the fourth week of class; and (3) Questionnaire. The module sections are further divided into 3 parts (a) Construction; (b) Compelling Argument; and (c) Confirmation.
Thematic Problems (Forms A and B). In order to make this exam more cohesive for you (and to provide you with
some learning options), I've constructed two distinct class modules and two distinct project modules (Forms A and
B). You'll need to turn in one class module of your choice and one project module of your choice.
Each module consists of 3 items: (3 items / module) x (12 points / item) x 2 modules = 72 points total). In
addition, you are to complete a short questionnaire item at the end of the exam (3 points).
Electronic Files. Several exam items require you to construct dynamic sketches with software. Create a new
directory in your MTH 606 Wiki "Upload Manager" space. Name this directory "Final_Exam." I will be looking for
this folder as I grade your exams.
Final Exam, MTH 606 (SU 09) 2
Section 1: Class Modules (FORM A)
Part 1: Construction
Construct an arbitrary quadrilateral ABCD with similar triangles EBA, HBC, GDC, and FDA constructed on the sides.
In your construction, triangles alternate - every other triangle lies in the interior of ABCD (see below).
Your sketch should be dynamic. When point E is dragged, points F, G, and H should move accordingly. I've
created a sample sketch for your inspection at the following link: http://tinyurl.com/examquad. Although you
can't save my file, you may use it for reference purposes as you create your own sketch.
Save your final sketch as Quad_Similar_Triangles.ggb in your MTH 606 Final_Exam folder.
Part 2: Compelling Argument
Connect the centroids of similar triangles EBA, HBC, GDC, and FDA to form a quadrilateral.
(a) What specific type of quadrilateral appears to be formed (e.g. square, kite, trapezoid)? In the textbox below,
provide a 1-2 sentence conjecture (2 pts.)
"Similar Triangles Quadrilateral" Conjecture 1:
(b) What specific geometric object appears to be formed by points E,F,G, and H? (hint: connect the points using the
polygon tool). In the textbox below, provide a 1-2 sentence conjecture (2 pts.)
"Similar Triangles Quadrilateral" Conjecture 2:
(c) Construct a compelling GeoGebra sketch that clearly illustrates the apparent relationships suggested in both
Conjectures 1 and 2 above (8 pts.). Save this sketch as SimilarTriangleQuad.ggb in your MTH 606
Final_Exam Folder.
Final Exam, MTH 606 (SU 09) 3
Part 3: Confirmation (Proof)
Compose a rigorous mathematical proof confirming either "Similar Triangles Quadrilateral Conjecture 1" or
"Similar Triangles Quadrilateral Conjecture 2." Write out all steps clearly, using proper mathematical
terminology in all cases. You are encouraged to include illustrations and labels to enhance the clarity of your
argument.
Final Exam, MTH 606 (SU 09) 4
Section 2: Project Modules (FORM A)
Part 1: Construction
Construct a GeoGebra custom tool that allows the user to click on three vertices of an arbitrary triangle ABC.
The tool generates a second triangle, XYZ, such that XY, YZ and XZ are the lengths of the medians of triangle
ABC. We'll informally refer to XYZ as the "median triangle" of ABC.
I've created a sample tool for your consideration. It is available for download at the following link (download
the tool, then add it to an existing sketch):
http://www.geogebra.org/en/upload/index.php?action=downloadfile&filename=MedianTriangle.ggt&directory=M
TH606/Todd_Edwards/Tools_Folder&
You are encouraged to experiment with the tool as you consider your own implementation. In the space
provided below, briefly describe steps you took to create your own tool.
Save your final tool as MedianTriangleTool.ggt in your MTH 606 Final_Exam folder.
Final Exam, MTH 606 (SU 09) 5
Part 2: Compelling Argument
A relationship holds between the area of an arbitrary triangle ABC and its "median triangle."
(a) What is this relationship? In the textbox below, provide a 1-2 sentence conjecture describing the apparent
relationship (2 pts.)
"Median Triangle" Conjecture:
(b) Construct a compelling GeoGebra sketch that clearly illustrates this apparent relationship (10 pts.). Save this
sketch as MedianTriangleSketch.ggb in your MTH 606 Final_Exam Folder.
Part 3: Confirmation (Proof)
Compose a rigorous mathematical proof confirming the "Median Triangle" conjecture. Write out all steps clearly,
using proper mathematical terminology in all cases. You are encouraged to include illustrations and labels to
enhance the clarity of your arguments.
Final Exam, MTH 606 (SU 09) 6
Section 1: Class Modules (FORM B)
Part 1: Construction
Construct an arbitrary triangle ABC. Then construct an equilateral triangle on each side of ABC making sure that
the three new triangles are on the "outside" of ABC. Label the vertices so that the new triangles are A'BC, AB'C, and
ABC'.
Construct the three centroids of triangles A'BC, AB'C, and ABC' and label them U, V, and W, respectively. Your
sketch should be dynamic. When points A, B, or C are dragged, triangles A'BC, AB'C and ABC' should remain
outside ABC. Points U, V, and W should remain centroids.
Save your final sketch as Equilateral_Triangles.ggb in your MTH 606 Final_Exam folder.
Part 2: Compelling Argument
Connect the centroids of triangles A'BC, AB'C, and ABC' to form triangle UVW.
(a) What specific type of triangle appears to be formed? In the textbox below, provide a 1-2 sentence conjecture
(2 pts.)
"Centroid Triangle" Conjecture 1: When equilateral triangles are constructed on the sides of an arbitray
triangle and then centroids of these equilateral triangles are constructed, another equilateral triangle is
formed by connecting the centroids.
(b) Next, construct lines AU, BV and CW. What relationship do these lines appear to share? In the textbox below,
provide a 1-2 sentence conjecture (2 pts.)
"Centroid Triangle" Conjecture 2: When equilateral triangles are constructed on the sides of an arbitray
triangle, centroids of these equilateral triangles are constructed, and then the segments connecting
vertices with opposing centroids are created, these segments are concurrent.
(c) Construct a compelling GeoGebra sketch that clearly illustrates the apparent relationships suggested in
Conjectures 1 and 2 above (8 pts.). Save this sketch as CentroidTriangle.ggb in your MTH 606 Final_Exam
Folder.
Final Exam, MTH 606 (SU 09) 7
Part 3: Confirmation (Proof)
Compose a rigorous mathematical proof confirming either "Centroid Triangle" Conjecture 1 or "Centroid
Triangle" Conjecture 2. Write out all steps clearly, using proper mathematical terminology in all cases. You are
encouraged to include illustrations and labels to enhance the clarity of your arguments.
I know that this point is called the Napolean point, I know the triangle is called the Napolean
triangle. Besides very difficult proofs, which I in no way understood, I could not find help to
construct a proof.
I tried looking at all six vertices and using either Pascal’s or Brianchon’s theorems, however the
six points were not either inscribed or circumscribed about a circle, so this was of no help. I tried
constructing parallel lines and finding similar triangles, I tried using DesArgue’s theorem. I tried
finding triangles that contained all six points on extended sides to use Menalause, but I am giving
up. Sorry.
Final Exam, MTH 606 (SU 09) 8
Section 2: Project Modules (FORM B)
Part 1: Construction
Given a triangle, extend two sides in the direction opposite their common vertex. The circle tangent to these two
lines and to the other side of the triangle is called an excircle. The center of the excircle is called the excenter.
Every triangle has three excircles,
Construct a GeoGebra custom tool that allows the user to click on three vertices of an arbitrary triangle ABC to
create an excircle. Specifically, when the user clicks on points A, B, and C in order with the tool, the following
objects are constructed:
(1) An excircle intersecting side AB
(2) The point of intersection of AB and the aforementioned excircle
I've created a sample tool that performs such a task available at the following link (download the tool, then add it
to an existing sketch):
http://www.geogebra.org/en/upload/index.php?action=downloadfile&filename=excircle.ggt&directory=MTH606
/Todd_Edwards/Tools_Folder&
Verify that the tool works by constructing a triangle ABC, using the tool to create excircles, then extending sides of
ABC to confirm tangent relationships. Use the tool as you consider your own implementation. In the space
provided below, briefly describe steps you took to create your own tool.
I created triangle ABC, then the lines extending the sides of the triangle. Next, I found the interior
angle bisector of the vertex opposite the side I wanted the excircle to be tangent to and the exterior
angle bisector of one of the vertices of the side I wanted the excircle to be tangent to. I found the
intersection of these lines, then found the perpendicular line from this point to the segment of the
triangle tangent to the excircle. Lastly, I found the intersection of the triangle segment and the
perpendicular line and created the circle with the radius at the intersection of the angle bisectors and
the point on the circle at the intersection of the perpendicular and the triangle segment.
Final Exam, MTH 606 (SU 09) 9
Save your final tool as ExcircleTool.ggt in your MTH 606 Final_Exam folder.
Part 2: Compelling Argument
Construct the three excircles for arbitrary triangle ABC. Label the point at which an excircle intersects side BC as
A'. Similarly label the point at which an excircle intersects side AC as B'. Lastly, label the point at which an excircle
intersects side AB as C'.
A relationship exists between lines AA', BB' and CC'.
(a) What is this relationship? In the textbox below, provide a 1-2 sentence conjecture describing the apparent
relationship (2 pts.)
"Excircles Vertex" Conjecture: The lines conecting the vertices of a triangle to the opposing points, which
are intersections of the excircle tangent to the triangle and the side of the triangle, are concurrent. Lines
AA’, BB’, and CC’ are concurrent.
(b) Construct a compelling GeoGebra sketch that clearly illustrates this apparent relationship (10 pts.). Save this
sketch as ExcirclesVertex.ggb in your MTH 606 Final_Exam Folder.
Part 3: Confirmation (Proof)
Compose a rigorous mathematical proof confirming the "Excircles Vertex" conjecture (hint: I've included a Nagel
Point tool for you in my Geogebra Tools Folder). Write out all steps clearly, using proper mathematical
terminology in all cases. You are encouraged to include illustrations and labels to enhance the clarity of your
arguments.
Based on the Excircle Theorems that states that the segment from a vertex of a triangle the adjacent
excircle tangent point is equal to the semiperimeter minus the adjacent side of a triangle:
AC’ = s – AC
C’B = s – BC
BA’ = s – AB A’C = s – AC
CB’ = s – BC
By Ceva’s Theorem (since segments AA’, BB’, and CC’ are cevians),
then through substitution,
property,
B’A = s – AB
AC ' BA ' CB'


should = 1
C ' B A' C B ' A
s  AC s  AB s  BC


should also =1; and through the commutative
s  BC s  AC s  AB
s  AC s  AB s  BC


 111
s  AC s  AB s  BC
Therefore, since the products of the proportions of the side lengths, cut by cevians is congruent to one,
then these cevians are concurrent.
Final Exam, MTH 606 (SU 09) 10
Section 3: Questionnaire (3 points possible - 1 point each)
(a) Recall that this exam is scored out of 75 possible points. Based on your perceptions of your own performance
today along with your sense of my grading style, predict your final score on this exam. Your prediction should
range somewhere between 0 and 75 points (inclusive).
Predicted Score: ____63/75___84%____________ (1 pt.)
(b) Based on your prediction, what letter grade do you anticipate receiving in MTH 606? ___92% or A-______ (1 pt.)
Comments (Optional)
(c) Please consider the following question from a philosophical point of view: "Did you see the gorilla during our
exam?" Please comment in several complete sentences. Creativity and humor are much appreciated (1 pt.)
To me the gorilla in the exam was the depth of the problems. I felt that I got too far into a problem and then
could no longer have time to switch back to the other problem when I had difficulty.
I TRULY HOPE YOU ENJOYED CLASS, LEARNED SOME NEW CONTENT, AND THOUGHT A BIT ABOUT HOW YOU
MIGHT USE TECHNOLOGY IN YOUR INSTRUCTION DURING THE UPCOMING YEAR. I TRULY ENJOYED SPENDING
TIME WITH YOU.
PLEASE CONSIDER ME A RESOURCE / COLLEAGUE / SYMPATHETIC EAR DURING THE SCHOOL YEAR. DON'T
HESITATE TO CALL ON ME IF YOU NEED ANYTHING.