Math 3395
Koppelman
HW # 12
Name__________________________
In this assignment, you are asked to use Geometer’s Sketchpad to make several
conjectures. Prove only those conjectures that specifically ask for proof. Please include
hard-copies of the constructions when you turn in this assignment.
B
1. In the last homework assignment, you constructed and saved the
1
1
1
diagram shown, in which AP = AB, CR = AC, and BQ = BC.
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3
3
P
a.
b.
Display the area of ABC and KLM
(the shaded triangle).
L
Q
K
A
M
Make a conjecture about how many times larger the area
ABC is than the area of KLM.
R
C
2. a. Using Geometer’s Sketchpad, construct a regular octagon.
b. Construct the midpoint of the longest diagonal and call it point P (as shown).
c. Using point P as center, construct a circle that passes through each vertex of the
octagon. We say that the octagon is inscribed in the circle.
d. Construct a segment ( PR ) from the circle’s center perpendicular to a side of the
octagon. Note, PR is called an apothem of the regular hexagon.
e. Display the area of the octagon.
f. By displaying the appropriate measurements, verify that
the area of the octagon is equal to one-half the product of
the apothem and the perimeter.
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1
(i.e. Area = (apothem)(perimeter) or A = ap )
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2
P
R
3. Using Geometer’s Sketchpad, construct a rhombus and its two diagonals.
a.
Display the lengths of both diagonals
b.
Display the area of the rhombus.
c.
Conjecture a formula for the area of a rhombus in terms of the product of the
lengths of the diagonals.
d.
Prove your conjecture.
4. Experiment on Geometer’s Sketchpad to find a way to obtain the area of
a. a circle
segment
sector
b. a sector of a circle.
c. a segment of a circle.
Find the area of each figure indicated below (questions 5-7):
5. ABC
6. Parallelogram EFGH
B
F
13
8.5
A 4
11
15
E
C
9
7. Isosceles trapezoid JKLM
J
G
14
H
8. In the parallelogram from question 6,
what is the length of the altitude from
H to FG ?
K
F
G
14
7
13
M
4
N
?
L
E
H
15
9. In the diagram, ABCD is a rectangle.
Point P is chosen randomly on side DC .
Conjecture a relationship between the area
of the rectangle and the area of APB (shaded).
Prove that your conjecture is true. (Note: use of
Sketchpad on this question is optional.)
D
P
A
10. Use the Geometer’s Sketchpad construction of the 13-14-15 triangle (sent to you earlier)
to complete this question.
a. Use the area option on Geometer’s Sketchpad to obtain the area of triangle ABC.
b. Construct an altitude from C to AB, measure its length, and use the formula A = ½bh
to verify the area you found in part (a).
c. Heron’s formula for the area of a triangle states that the area of triangle with side
lengths a, b, and c can be obtained using the formula Area = S (S  a)( S  b)( S  c)
where S is half the perimeter of the triangle (S is called the semi-perimeter, and
Heron’s formula is often referred to as the semi-perimeter formula). Use Heron’s
formula to verify the area you obtained in parts (a) and (b).
11. Use Geometer’s Sketchpad to construct a parallelogram whose area does not change
when its vertices are dragged. (Hint: look back at question 9.)
C
B