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Unformatted text preview: Math131 FINAL Name A
1. A rotation has been performed that sends triangle A to triangle B. Find the center of the rotation (any method). Wm 2. State the Pythagorean theorem. 3. Give a proof of the Pythagorean theorem. You may use any proof you Wish. You may want to base your proof on the ﬁgure below. [0 4a. What rigid motion sends A into B? Circle the correct one. Translation Rotation Reﬂection Slideﬂip. A 4b.What rigid motion sends A into B? Circle the correct one. Translation Rotation Reﬂection Slide—ﬂip. A 5. Refer to the picture below. If we reﬂe net effect is a rotation. ct about L1, and reﬂect the result about L2, the Locate the center of this rotation If we View the rotation as clockw ise, then the angle of rotation is If we View the rotation as counterclockwise, then the angle of rotation is L2 145 6. If a convex polyhedr on has 21 edges and 10 faces then how many ve 35 L1 rtices does it have? Answer: . triangle B. Label all lines. , . . ., such that the reﬂection
' iangle A to lden rectangle using straight—edge and compass only. IWhere does the 8. Construct a go
rightmost edge go? 9. Suppose a given regular polygon has a vertex angle with ~measure 174 degrees. (a) How many vertices does this polygon have? (b) What is its central angle? 10. Suppose n m 11 12 13 14 15 16 17 18 19 2O 21 22 23 24 25 26 11. How many distinct star polygons are there which have 22 vertices? CLTLS : 12. Circle the numbers below that are Fermat primes. 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 13. Can a regular 340—
your answer. — 22. Circle the integers d below for which the star polygon { gon be constructed with compass and straightedge onl y? Explain 14. With straight~edge and compass alone, construct a regular S—gon. Show your work. 3 correspond to a semi—regular tesselation? For
perfectly around a vertex, mine if the conﬁguration 15. Which of the following vertex sequence
each of the ﬁve cases below, determine if the required polygons ﬁt with no overlapping or missing space. If this can be done deter
can be extended to a semiregular tesselation. Vertex sequence Perfect ﬁt around a vertex? Extendable? (if applicable) (3,3,6,6) Y N Y N
(4,12,12) Y N Y N
(4,6,12) Y N Y N
(3,6,3,6) Y N Y N
(3,4,3,3,4) Y N Y N 16. For the polygon below, ﬁnd the sum of the vertex angles. Sum is f 17. Sketch the Schlegel diagram for the icosahedron. 10 umber of axes of rotational symmetry of order 3 is equal to 18a. For the tetrahedron, the n ______’______‘——————— and the number of planes of reflectional symmetry is equal to . 18b. For the octahedron, the number of axes of rotational symmetry of order 3 is equal to and the number of planes of reﬁectional symmetry is equal to . 18c. For the icosahedron, the number of axes of rotational symmetry of order 2 is equal to f and the number of planes of reﬂectional symmetry is equal to .
19. Consider the dodecahedron. The number of faces is The number of edges is The number of vertices is What does Eulers formula say about these three quantities? 11 ge and compass alone ﬁnd the circumcenter, the centroid, the or— 20. Using a straighted
thocenter and the Euler line. Label your results. 12 21. Apply a 90 degree clockwise rotation about the centerpoint a: to the triangle A below,
to get a triangle A’. Apply a 90 degree clockwise rotation about the centerpoint y to A’,
to get a triangle A”. 'What single rigid motion sends A to‘A”? Describe this rigid motion completely.
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 Fall '07
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