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Week 6 Test Please Complete Without Input from Others Show calculations to earn full credit. Calculations are worth 50% points 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.