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Unformatted text preview: Math131 FINAL (MWF 12:05PM Section) Fall 2006 Name 1. (Straightedge and compass construction) Find the reﬂection of triangle A about the
given line. RN 2. By folding this page, ﬁnd at most 3 lines of reﬂection L1, L2, . . . such that the reﬂection V
about L1, followed by the reﬂection about L2, etc. has the net effect of sending triangle A
to triangle B. Label all lines. 3. Apply a 90 degree counterclockwise rotation about the centerpoint z to the triangle
A below, to get a triangle A’. Apply a. 90 degree counterclockwise 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. Ans 0 o O O o 0 Q o 6 0 o g
0 g g g o o o o o g o .
O O O a O o o o t c. . .
O U C o O . . . 5 ' . .
° 0 o ' O a 3 0 O 4 o L
o o I o c , . ' l o a .
' a o a I o .0 c a o o a
a q I a o v 9 .X o 0 0 o
O O I O p g g , . . ' . a a y C o o o q a
Y O . a g . . ' . . ‘ 6 o 0 g g . ' . C d f O I I o . . . O . 4a. What rigid motion sends A into B? Circle the correct one. Translation Rotation Reﬂection Slide—ﬂip. A 4b. What rigid motion sends A into B? Circle the correct one. Tianslation Rotation Reﬂection Slideﬂip. A 5. Reﬂect triangle A aboutthe line L1 to get a triangle A’. Reﬂect triangle A’ about the
line L2 to get a triangle A". What single rigid motion sends A to A”? Describe this rigid
motion completely. Ans 6. Consider the translation that sends triangle, A to triangle B. Draw an arrow that
represents this translation. Assuming that adjacent grid points below are one centimeter apart, ﬁnd the distance of this slide. 0 § 0 o o . Q . . O .
' ' I o 9 Q o g
o . o o a o p o
0» O o O O t ¢ 0
o O a . g ‘ . 5 ' .
0 Q o v 0 a, 3 0 o a o
o d a o c . g ' ‘ o o
0 a o u I 0 0 0 O o a
5 0 v q o u . o . , .
o a n 0 a a a .
I 0 y i 0 q 0 o
l I u q 0 .  a
6 a Q . a . :4 C I g g
. ‘ ¢ 0 p a o . Q g , 7. If a convex polyhedron has 22 edges and 10 faces then how many vertices does it have? Answer: 8. In the space below, sketch a 6—gon that is simple but not convex. In this polygon there
are six vertex angles. What is the sum of the measures of these six angles? 9. In the space below, sketch a convex 6gon that has order 3 rotational symmetry ,but no
lines of reﬁectional symmetry. ' 10. In the space below, sketch anexample of a star polygon that has 8 vertices. 11. State the Pythagorean theorem. 12. 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. 13. Construct a golden rectangle using straightedge and compass only. Where does the
rightmost edge go? ' 14. Suppose a given regular polygon has a vertex angle with measure 170 degrees.
(a) How many vertices does this polygon have? (b) What is its central angle? 10 15. Suppose n = 24. Circle the integers at below for which the star polygon exists. 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 16. How many distinct star polygons are there which have 24 vertices? ans = 17. Circle the numbers below that are Fermat primes. 12 13 14 15 16 17 18 19 20 21 22 23 24 25 18. Can a regular 680~gon be constructed with compass and straightedge only? Explain
your answer. ' 11 19. With straight—edge and compass alone, construct a regular 5g0n. Show your work. 12 20. Which of the following vertex sequences correspond to a semi—regular tesselation? For
each of the ﬁve cases below, determine if the required polygons ﬁt perfectly around a vertex,
with no overlapping or missing space. If this can be done determine if the conﬁguration
can be extended to a semiregular tesselation. Vertex sequence Perfect ﬁt around a vertex? Extendable? (if applicable)
(4, 6, 12) Y N Y N
(4, 12, 12) Y N Y N
(3, 6, 3, 6) Y N Y N
(3, 3, 6, 6) Y N y N
(3, 4, 3, 3, 4) Y N Y N 13 21. Sketch the Schlege] diagram for the icosahedron. 14 22a. For the dodecahedron, the number of axes of rotational symmetry of order 2 is equal
to and the number of planes of reﬂectional symmetry is equal to 22b. For the tetrahedron, the number of axes of rotational symmetry of order 3 is equal
to and the number of planes of reﬂectional symmetry is equal to 220. For the octahedron, the number of axes of rotational symmetry of order 4 is equal to and the number of planes of reﬂectional symmetry is equal to 15 23. For the regular polygon below, ﬁnd the measures of the given angles. 16 24. Consider an equilateral triangle with side length 4\/§ inches. Find the distance from
each vertex to the circumcenter. 17 ...
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