MATH 3395 GROUP PROBLEM SOLVING PROJECT
This Group Problem Solving Project has two parts.
Part I - consists of ten (10) challenging geometry
problems. Each requires that a conjecture be
made and/or verified by using Geometer’s
Sketchpad, and all but one require proof of the
conjecture. In order to receive full credit, your
group must complete six (6) of the problems.
Part II – An article taken from the Mathematics
Teacher (a publication of the National Council of
Teachers of Mathematics). In order to receive full
credit, your group must read the article and answer
the questions relating to it. These questions
require the use of Geometer’s Sketchpad.
This project is worth a total of 70 points.
PART I
For each problem in Part I, your group must submit:
Electronically
1. The Geometer’s Sketchpad screen(s) that led to
your conjecture. These should include a display of
relevant measurements. Make sure that the
information given in the problem remains unchanged
when the size of the sketch is changed by dragging
points. The sketches may either be transferred
from your flash-drive to mine in class or attached to
email messages and sent to ckoppelm@kennesaw.edu.
The problem number and the names of the group
members must be included in the Sketchpad screen.
On Paper
2. A clear statement of your group’s conjecture (for
those problems that ask you to formulate a
conjecture).
3. A clearly presented proof of the conjecture.
I. The Angle Bisector Problem
In the diagram below, triangle ABC is an acute, scalene triangle with side AC
lying along ray AD. Ray AP bisects angle BAC and ray CP bisects angle BCD.
1.
Construct the diagram using Geometer’s Sketchpad.
2.
Make a conjecture about how the measures of angle ABC and angle APC
compare.
3.
Drag point B and verify your conjecture (or form a new conjecture if your
first conjecture turns out to be incorrect).
4.
When you are confident that your conjecture is correct, prove it.
P
B
A
C
D
II. An Unexpected Result
1. Use Geometer’s Sketchpad to construct a parallelogram.
2. Construct the angle bisectors of each of the angles of the
parallelogram.
3. The points of intersection of each pair of angle bisectors are the
vertices of a quadrilateral. Make a conjecture about what type of
quadrilateral it is. Be as specific as possible.
4. Prove your conjecture.
III. Star Polygons
1.
Use Geometer’s Sketchpad to make a conjecture about the sum of the
measures of the five angles at the “points” of a five-pointed star.
2.
Prove that your conjecture is true.
C
B
D
A
E
IV. Two Interesting Triangles
1. Use Geometer’s Sketchpad to construct a triangle whose sides have
lengths in the ratio of 4 : 5 : 6.
2. Measure the smallest angle and the largest angle of the triangle.
3. Make a conjecture about how the measures of the two angles
compare.
4. Use Geometer’s Sketchpad to construct a triangle whose sides have
lengths in the ratio of 3 : 8 : 10.
5. Measure the two largest angles of the triangle.
6. Make a conjecture about how the measures of the two angles
compare.
7. Prove either one of the two conjectures above.
V. Twin Poles
Two flag poles of different heights are placed on level ground and held in
vertical position by wires connecting the top of each to the bottom of the
other (as illustrated in the diagram below).
Wires
KSU
Pole 1
Q
Pole 2
1.
Simulate the situation with a Geometer’s Sketchpad construction.
(You do not need to include the flags.)
2.
Using Geometer’s Sketchpad, display the ratio of the product of the
heights of the two poles to the sum of the heights of the two poles.
3.
Measure and display the height above the ground of point Q.
4.
Make a conjecture about how the ratio in question 2 is related to the
height in question 3.
5.
Prove your conjecture.
VI. A Point About Rectangles
Point P is chosen inside rectangle ABCD and
line segments PA, PB, PC, and PD are drawn.
B
A
1. Construct the diagram using
Geometer’s Sketchpad.

P
C
D
2. Make a conjecture about how the squares of the
lengths of PA, PB, PC, and PD are related.
3. Drag point P to a new location within the rectangle to
test whether your conjecture is still valid.
4. If you are confident that your conjecture is correct,
prove it.
VII. An Unexpected Ratio
In right triangle ABC, leg AC is twice the
length of leg AB. From the midpoint M of
AB, a perpendicular is drawn to hypotenuse BC.
C
1. Construct the diagram with Geometer’s
Sketchpad.
2. Make a conjecture about the ratio of
BC : BD.
D
3. Prove your conjecture.
A
M
B
VIII.
Right Triangles and Inscribed Circles
1. Use Geometer’s Sketchpad to construct the diagram below, in which
 Triangle ABC is a right triangle, with right angle at point C
 A circle inscribed in triangle ABC.
2. Using Geometer’s Sketchpad, verify that the radius of the inscribed
circle is equal to twice the area of the triangle divided by the
perimeter of the triangle.
A
3. Prove that this relationship is true for
the inscribed circle in any right triangle.
B
C
IX.
Fun with a Regular Nonagon
1. Use Geometer’s Sketchpad to construct the regular nonagon below with
diagonals AB and AC.
2. Make a conjecture about how the length of the side of the regular
nonagon is related to the lengths of AB and AC.
3. Prove your conjecture.
A
C
B
X. Where Does Harry Potter Buy His School Supplies?
In the graph below, the coordinates of the points shown are A(1,6), B(2,2),
C(8,1), D(7,5), and P(6,8).
1. Construct the diagram using the graphing options in Geometer’s Sketchpad.
2. Construct a line that passes through point P and divides parallelogram ABCD
into two quadrilaterals with equal areas. Display the areas of the two
quadrilaterals.
3. Choose another point outside parallelogram ABCD and label it Q. Construct a
line through Q that also divides the parallelogram into two quadrilaterals with
equal areas.
4. The two lines you constructed intersect in a point. Make a conjecture about
how this point can be precisely located.
5. Prove that any line drawn through the point you identified with your conjecture
divides a parallelogram into two regions with equal areas.
P
8
6
A
D
4
2
B
C
5
-2
10
PART II – NCTM ARTICLE
Extending a Classical Geometry Problem
In this article, an interesting lemma about isosceles triangles is examined.
The lemma says that for any point on the base of an isosceles triangle, the
sum of the distances from the point to the legs of the triangle is constant.
The lemma is then extended to apply to any point P chosen on the line
containing the triangle’s base. Special cases are also examined.
To receive full credit, your group must:
a. Use Geometer’s Sketchpad to construct Figures 1 and 2a
b. Make sure the triangles remain isosceles when points are dragged to
change their sizes and angle measures.
c. For each figure, display the individual distances from point P to the legs,
and their sum.
d. Use Geometer’s Sketchpad to verify that the formula shown in Figure 1
in the article (PQ + PR = AB∙sin A) is correct.
e. Use Geometer’s Sketchpad to construct the diagram in Figure 3 in
the article.
f. Use Geometer’s Sketchpad to verify that the caption under Figure 3
is correct.
g. Prove that the caption under Figure 3 is correct for all equilateral
triangles.
h. Email the Geometer’s Sketchpad constructions to me or transfer them
from your flash-drive to mine.
i. Include a hard copy of the Sketchpad screens and your proof when you
submit your project.
From: The Mathematics Teacher
April 2009
Volume 102 #8