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Week 6 Test
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Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or
answers the question.
Provide an appropriate response.
1. F25 = 75,025 and F26 = 121,393 where Fn is the nth term in the Fibonacci sequence. Find F27.
A) 242,786
B) 196,418
C) 46,368
D) 167,761
F27 = F26+F25 by the way the Fibonacci sequence is defined. So you just have to add the given
numbers.
2. If an 8-inch wide rectangle is to approach the golden ratio, what should its length be?
A) 16 in.
B) 13 in.
C) 10 in.
D) 12 in.
It is known that the ratio Fn+1/Fn approaches the golden ration when n is large. Since the
given width is 8=F6,then the length should be F7=13.
Find the area of the specified figure (Q3 and 4).
3. Rectangle ABCD
A) 24 square units
B) 42 square units
C) 48 square units
D) 21 square units
4. Find the area of the specified figure
ABC
A) 20 square units
B) 31 square units
C) 40 square units
D) 80 square units
Height of the triangle in 8 units, base of the triangle is 10 units. Area = base x height/2 = 40.
5. Which basic geometric idea is suggested by a spoke in a bicycle wheel: a ray, a line, a line
segment, or an angle?
A) Ray
B) Line
C) Line segment
D) Angle
6. Which descriptor could be used to characterize three lines that intersect at the corner of a cube:
coplanar, cubic, concurrent, or complementary?
A) Coplanar
B) Cubic
C) Concurrent
D) Complementary
7. Find a traversable path that begins at vertex A.
A) A → C → D → A → B → D
B) B → A → D → C
C) A → B → D → C
D) No such path exists.
8. What is the name for the point where the three altitudes of a triangle are concurrent?
A) centroid
B) circumcenter
C) orthocenter
D) incenter
9. Which point of concurrency in a triangle is the center of a circle that contains the vertices of the
triangle?
A) orthocenter
B) incenter
C) centroid
D) circumcenter
10. What is the angle sum of a regular polygon with 7 sides?
A) 900°
B) 1350°
C) 450°
D) 1260°
The sum of the angles of a heptagon is 5x180= 900. To see this do the following thing. Pick a
point inside the heptagon and join it to all the 7 vertices. The heptagon is now split into 7
triangles.
The sum of the angles of all these triangles is 7x180. But to get only the sum angles of the
heptagon you have to subtract the angles formed around the interior point which add up to
360. So the answer is 7x180-360= 900.
A different way: join a vertex of the heptagon with all the other vertices by drawing
diagonals. The heptagon is now split into 5 triangles; the sum of the angles of these triangles
matches the sum of the angles of the heptagon, that is 5x180=900.
11. If each angle of a regular polygon measures 140°, how many sides does it have?
A) 11 sides
B) 9 sides
C) 7 sides
D) 3 sides
Suppose the polygon has n sides. Such a polygon can be split into n-2 triangles by drawing all
the diagonals from one vertex. It follows that the sum of the angles of the polygon is (n2)x180. Since the polygon is regular, all n angles are congruent to each other; so each angle
measures (n-2)x180/n. Setting this equal to 140 we get (n-2)/n =140/180 = 7/9 from which
n=9.
12. What type of figure is formed by joining the midpoints of the sides of a general quadilateral?
(Be as specific as possible.)
A) parallelogram
B) rhombus
C) square
D) rectangle
13. Choose the word "rectangle," "square," or "rhombus" to fill in the blank and make the
statement true.
(If none of the words can be used, then write "none.")
A parallelogram is a _________ if and only if its diagonals are congruent.
A) rhombus
B) square
C) rectangle
D) none
14. What is the angle sum of a regular polygon with 4 sides?
A) 540°
B) 360°
C) 180°
D) 720°
A regular polygon with 4 sides is a square. A square has 4 angles of 90 degrees each so 360
degrees in total.
15. If each angle of a regular polygon measures 108°, how many sides does it have?
A) 5 sides
B) 3 sides
C) 4 sides
D) 7 sides
Same approach as in 11. Suppose the polygon has n sides. Such a polygon can be split into
n-2 triangles by drawing all the diagonals from one vertex. It follows that the sum of the
angles of the polygon is (n-2)x180. Since the polygon is regular, all n angles are congruent
to each other; so each angle measures (n-2)x180/n. Setting this equal to 108 we get (n-2)/n
=108/180 = 3/5 from which n=5.
Determine if the given regular figures form semiregular tessellations.
16. Square, two octagons
A) Yes (see figure 11.18 h)
B) No
Determine if the given regular figures form semiregular tessellations
17. Two triangles, two hexagons
A) No
B) Yes (see figure 11.18 e)
Answer the question.
18. Will any parallelogram tessellate the plane?
A) yes
B) no
19. What must be true for a pentagon so that it will tessellate a plane?
A) It must be a regular pentagon.
B) It has one pair of parallel sides.
C) It never tessellates a plane.
D) All of its sides must be the same length.
Here things are a bit unclear to me; first A) and C) are definitely false.
The answer is either B or D. However, it may happen that we have tessellations
for pentagons which satisfy neither B nor D. For example, see tessellations 2, 3, 5 in the
list below.
http://upload.wikimedia.org/wikipedia/commons/1/1e/PentagonTilings.svg
So, I would ask the instructor what was the intent of this question.
Use Euler's formula to answer the question.
20. A polyhedron has 13 vertices and 21 faces. How many edges does it have?
A) 34
B) 32
C) 36
D) 33
In every polyhedron we have the Euler formula:
#vertices+#faces=#edges+2. In your case: 13+21=#edges+2 from which
#edges=34-2=32.