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divided
alphabet
categories
I’veI’ve
divided
the the
alphabet
intointo
fivefive
categories
as as
shown
below.
identify
category
shown
below.
CanCan
youyou
identify
the the
category
to to
which
each
letter
of the
word
BRAIN
belongs?
which
each
letter
of the
word
BRAIN
belongs?
Give a brief explanation for your choice.
Find another five letter word whose letters
each belong in different categories.
(1) M T U V W Y
(2) C D E K
(3) S Z
(4) H O X
(5) F G J L P Q
Midterm Exam
24 students took the midterm
Mean = 120/150 = 80%
Median = 133.5/150 = 89%
Questions 3b and 3c refer to the third Sketchpad construction that you submitted
electronically. The diagram and information appear below.
D
C
E
ABCD is a rhombus
Point P is any point on diagonal AC that
is closer to point A than it is to point C
P
PF  AB
EP  AC
A
F
B
3b. A high school geometry student proved that the two smaller triangles in the diagram
are congruent using the following argument:
1. EAP  FAP (The diagonals of a rhombus bisect the angles of the rhombus.)
2. EPA  PFA (They are both congruent right angles.)
3. AP  AP (Reflexive postulate - both triangles have this side in common.)
3. The two triangles are congruent by AAS.
Is this proof valid? If not, explain why.
3c. The same student then concluded that AEP and APF are congruent. Do you
agree that these two angles are congruent? Explain your answer.
MATH 3395
Fall, 2014
MIDTERM
E
87
7. In the diagram, AE is parallel to JB . What is the measure
of J? Briefly explain how you obtained your answer.
J
62
A
B
C
D
MATH 3395
Fall, 2014
MIDTERM
10. In the diagram, two pairs of segments are parallel,
as indicated. What is the sum of the measures of
1, 2, and 3?
1
43
2
3
A
11. In the diagram, M is the midpoint of BC and
AME is a right angle. What is the sum of the
measures of A, B, C, D, and E?
Explain how you obtained your answer.
E
D
B
M
12. The measure of one of the angles of a rhombus is 60 and the length of a side is
10 inches. What is the length of the shorter diagonal of the rhombus? Draw a
diagram and explain how you obtained your answer.
C
AME is a right angle. What is the sum of the
measures of A, B, C, D, and E?
Explain how you obtained your answer.
E
D
B
M
12. The measure of one of the angles of a rhombus is 60 and the length of a side is
10 inches. What is the length of the shorter diagonal of the rhombus? Draw a
diagram and explain how you obtained your answer.
C
In each diagram below (questions 15 – 18), information is marked. State whether two
triangles in each diagram can be proven congruent based on the information given and
valid deductions that can be made from it. If they can be proven congruent, state the
congruence theorem that applies.
15.
16.
70
110
f) True or False: The measure of DEC
is equal to the sum of the measures of
DAC and DCE. Justify your answer.
divided
alphabet
categories
I’veI’ve
divided
the the
alphabet
intointo
fivefive
categories
as as
shown
below.
identify
category
shown
below.
CanCan
youyou
identify
the the
category
to to
which
each
letter
of the
word
BRAIN
belongs?
which
each
letter
of the
word
BRAIN
belongs?
Give a brief explanation for your choice.
Find another five letter word whose letters
each belong in different categories.
(1) M T U V W Y A
(2) C D E K B
(3) S Z N
Vertical line symmetry
Horizontal line symmetry
180° rotational symmetry
(4) H O X I
Horizontal and vertical line symmetry,
and 180° rotational symmetry
(5) F G J L P Q R
No symmetry
Transformations
• Reflections
• Rotations
These transformations are isometries.
• Translations
• Dilations
An isometry is a transformation in which the
original figure and its image are congruent.
Thus isometries preserve distance and angle measure.
Reflections
If A is the reflection image of point A over line l, then
l is the perpendicular bisector of line segment AA
A
l
A
The Burning House Problem
A man is walking in an open field some distance from his house. It’s a
beautiful day and he is carrying an empty bucket with him to collect berries.
Before long, he turns around and, to his horror, sees that his house in on fire.
Without wasting a moment, he runs to a nearby river (which runs in a straight
line from east to west) to fill the bucket with water so that he can run to his
house to throw water on the fire. Naturally, he wants to do this as quickly as
possible. Describe how to construct the point on the river bank to which
he should run in order to minimize his total running distance (and time).
H

M



M
H

M



M

Q
On a billiard table, a player must hit ball A into pocket B without
touching the other three balls shown. Where should the player aim?
B
Reflection image of point A.
B
B
Reflection image of point B.
B
B
Using Geometer’s Sketchpad, construct a rhombus
(not a square) by using only the 2 basic draw tools
(point and line segment) and reflections. The figure
should remain a rhombus when points are dragged.
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.
Regular Polygon – A polygon with all sides congruent and all angles congruent.
Sum of the interior angles = (n  2)180
Measure of each interior angle =
(n  2)180
n
Every regular polygon can
be inscribed in a circle.
Measure of each central angle =
Number of diagonals =
n(n  3)
2
360
n
140
40
Rotations
Use Geometer’s Sketchpad to construct a regular nonagon.
Rotations
Use Geometer’s Sketchpad to construct an angle ABC near the left side of
the screen. Construct a line segment DE somewhere else on the screen.
Use Sketchpad to construct an angle congruent to BAC and having
segment DE as one of its sides.
B
E
C
A
D
Translations
- Often called glide or slide transformations
Use Geometer’s Sketchpad to construct an acute triangle ABC near the
lower left of the screen. Construct point P in the center of your screen.
Use Sketchpad to construct a triangle congruent to ABC and having
point P as one of its vertices.
P
B
C
A
If you don’t complete these in class, complete as part of tonight’s homework.
HW problem 1:
Use Geometer’s Sketchpad to construct a rectangle whose
side lengths are in the ratio of 2:1 without using the
perpendicular, parallel, or midpoint options in the construct
menu, and without constructing any circles.
HW Problem 2:
Construct the letter
A using Geometer’s Sketchpad.
A
Your
must be perfectly vertical and symmetric. In other
words, it cannot look like this
or this
.
HW Problem 3:
Construct a line segment AB using Geometer’s Sketchpad.
Without changing line segment AB in any way, construct a
rhombus (not a square) so that AB is one diagonal of the
rhombus.