Difference between revisions of "1994 AHSME Problems/Problem 26"
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<math> \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 14 \qquad\textbf{(D)}\ 20 \qquad\textbf{(E)}\ 26 </math> | <math> \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 14 \qquad\textbf{(D)}\ 20 \qquad\textbf{(E)}\ 26 </math> | ||
==Solution== | ==Solution== | ||
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To find the number of sides on the regular polygons that surround the decagon, we can find the interior angles and work from there. Knowing that the measure of the interior angle of any regular polygon is <math>\frac{(n-2)*180}{n}</math>, the measure of the decagon's interior angle is <math>\frac{8*180}{10} = 144</math> degrees. | To find the number of sides on the regular polygons that surround the decagon, we can find the interior angles and work from there. Knowing that the measure of the interior angle of any regular polygon is <math>\frac{(n-2)*180}{n}</math>, the measure of the decagon's interior angle is <math>\frac{8*180}{10} = 144</math> degrees. | ||
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The regular polygons meet at every vertex such that the angle outside of the decagon is divided evenly in two. With this, we know that the angle of the regular polygon is <math>216/2=108</math> degrees. Using the previous formula, <math>n=5</math> <math>\boxed{\textbf{(A) }5}</math> | The regular polygons meet at every vertex such that the angle outside of the decagon is divided evenly in two. With this, we know that the angle of the regular polygon is <math>216/2=108</math> degrees. Using the previous formula, <math>n=5</math> <math>\boxed{\textbf{(A) }5}</math> | ||
| + | ==See Also== | ||
| − | - | + | {{AHSME box|year=1994|num-b=25|num-a=27}} |
| + | {{MAA Notice}} | ||
Latest revision as of 16:32, 9 January 2021
Problem
A regular polygon of 
sides is exactly enclosed (no overlaps, no gaps) by regular polygons of 
sides each. (Shown here for 
.) If 
, what is the value of ?
![[asy] size(200); defaultpen(linewidth(0.8)); draw(unitsquare); path p=(0,1)--(1,1)--(1+sqrt(2)/2,1+sqrt(2)/2)--(1+sqrt(2)/2,2+sqrt(2)/2)--(1,2+sqrt(2))--(0,2+sqrt(2))--(-sqrt(2)/2,2+sqrt(2)/2)--(-sqrt(2)/2,1+sqrt(2)/2)--cycle; draw(p); draw(shift((1+sqrt(2)/2,-sqrt(2)/2-1))*p); draw(shift((0,-2-sqrt(2)))*p); draw(shift((-1-sqrt(2)/2,-sqrt(2)/2-1))*p);[/asy]](http://latex.artofproblemsolving.com/7/2/d/72d4bad2396126c1291a75d31e70437f856c702a.png)
Solution
To find the number of sides on the regular polygons that surround the decagon, we can find the interior angles and work from there. Knowing that the measure of the interior angle of any regular polygon is 
, the measure of the decagon's interior angle is 
degrees.
The regular polygons meet at every vertex such that the angle outside of the decagon is divided evenly in two. With this, we know that the angle of the regular polygon is 
degrees. Using the previous formula, 
See Also
| 1994 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 25 |
Followed by Problem 27 | |
| 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 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
