MTH/SLS218-6816
Exam1
27,2008
February
ProfessorIlya Kofman
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Problem 1. Consideran equilateralAABC. A personbeginswalking straight at
point /, turns at B and then at C , and returns to A but does not turn. What is
the total number of degreesthat the walker has turned? Justify your answer.
l?-o+l?n= Ll O
Problem 2. (") If a pyramid hasa 100-gonfor its base,how many vertices,edges
and facesdoesit have? V = /eo tt
= /o I
lA
'--/-
e-
/ao+teo
{-/oo+/
= ?N
:/o/
(b) If a pyramid has 14 edges,how many facesdoesit have?
T-fro,L^f.L4 *= 7+t = g
(") Verify that your numbersin (a) and (b) satisfy Euler's formula.
a)
(")
)
V -' e f
= Z-
tgl -/,.oo +lo ( = L
>L
8'l'l +g
Problem 3. What fact about intersectingspheresenablesthree GPS satellitesto
determineyour exact location on Earth?
el
J
fil€f,eg i^ker*,*
i,,' /r-*
Po;^f9,
oho ,( U[ftI
ir o+1{*tl4-
Problem 4. What fact about the interior anglesof a regular octagonshowsthat
a regular polyhedroncannot have octagonalfaces?
int.
^#=
(p
(
ry
= (e# =t3f)
j*lgS=tlos>360
Problem 5. Considerthe earth and moon as shown.
(") rsthemoo"*€;,
(b)Irit
ft
tgpor
i,
\
waning?
o, ruu?
O
??
fr
Problem 6. For eachof the following pairs, identify which type of Venn diagram
describestheir relationship.
4 .l
l/ |
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oo
disjoint
overlapping
(u) Concaveshapesand polygons
subset
OVer(aP
5 t t &g o +
(b) Convex shapes and parallelograms
(.) Isoscelestriangles and right triangles
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(d) Rectanglesand regular quadrilaterals
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(") Platonic solidsand pyramids
(f) Prisms
andspheres
(s)Parallelograms and rhombi
(h) Kites and rhombi
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Oerl
L 9io;
--------_----_'-
a.P
^+
9q[ 9e-+
saI s2+
Problem 7. Among parallelograms, rectangles, rhombi, and isoscelestrapezoids,
for which ones are the following statements always true:
I
(") Diagonals
arethesamelength.
Recfon\A,
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h(-
RLo,.(i
7 (b)
Diagonalsbisectangles.
9 (")
Diagonals
crossat right angles. A L o ^ L i
I4
I u
iSos.b(es
Problem 8. Find the missinganglemeasuresindicated by letters in the diagram
below. Two parallel lines are indicated by arrows.
&2
9o
l-
5o
O--
c= fo
J=(oo
e=llo
+=lLo
5'U
Compass and straightedge constructions. Please do each one separately.
Problem 9. A
Given segmentAB, construct an equilateraltriangle with side AB.
Problem 10.
angle.
Draw an anglethat is approximately60". Preciselybisectthis
Problem 11. A
Given segmentAB, draw a point C above it. Construct a line parallel to AB
through C by copying an angle.
BONUS 12.
A
BC
Construct a rectanglewith given sidesAB and C D.
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