Difference between revisions of "2015 AMC 12B Problems/Problem 7"
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A regular 15-gon has <math>L</math> lines of symmetry, and the smallest positive angle for which it has rotational symmetry is <math>R</math> degrees. What is <math>L+R</math> ? | A regular 15-gon has <math>L</math> lines of symmetry, and the smallest positive angle for which it has rotational symmetry is <math>R</math> degrees. What is <math>L+R</math> ? | ||
| − | <math>\textbf{(A)}\; | + | <math>\textbf{(A)}\; 24 \qquad\textbf{(B)}\; 27 \qquad\textbf{(C)}\; 32 \qquad\textbf{(D)}\; 39 \qquad\textbf{(E)}\; 54</math> |
==Solution== | ==Solution== | ||
| − | + | From consideration of a smaller regular polygon with an odd number of sides (e.g. a pentagon), we see that the lines of symmetry go through a vertex of the polygon and bisect the opposite side. Hence <math>L=15</math>, the number of sides / vertices. The smallest angle for a rotational symmetry transforms one side into an adjacent side, hence <math>R = 360^\circ / 15 = 24^\circ</math>, the number of degrees between adjacent sides. Therefore the answer is <math>L + R = 15 + 24 = \boxed{\textbf{(D)} \, 39}</math>. | |
==See Also== | ==See Also== | ||
{{AMC12 box|year=2015|ab=B|num-a=8|num-b=6}} | {{AMC12 box|year=2015|ab=B|num-a=8|num-b=6}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Latest revision as of 01:27, 4 March 2015
Problem
A regular 15-gon has 
lines of symmetry, and the smallest positive angle for which it has rotational symmetry is 
degrees. What is 
?
Solution
From consideration of a smaller regular polygon with an odd number of sides (e.g. a pentagon), we see that the lines of symmetry go through a vertex of the polygon and bisect the opposite side. Hence 
, the number of sides / vertices. The smallest angle for a rotational symmetry transforms one side into an adjacent side, hence 
, the number of degrees between adjacent sides. Therefore the answer is 
.
See Also
| 2015 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 6 |
Followed by Problem 8 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
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