Difference between revisions of "1995 AHSME Problems/Problem 17"
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Given regular pentagon <math>ABCDE</math>, a circle can be drawn that is tangent to <math>\overline{DC}</math> at <math>D</math> and to <math>\overline{AB}</math> at <math>A</math>. The number of degrees in minor arc <math>AD</math> is | Given regular pentagon <math>ABCDE</math>, a circle can be drawn that is tangent to <math>\overline{DC}</math> at <math>D</math> and to <math>\overline{AB}</math> at <math>A</math>. The number of degrees in minor arc <math>AD</math> is | ||
− | [[ | + | <!-- original image at [[File:1995 AHSME num.17.png]] --> |
+ | <asy>size(100); defaultpen(linewidth(0.7)); | ||
+ | draw(rotate(18)*polygon(5)); | ||
+ | real x=0.6180339887; | ||
+ | draw(Circle((-x,0), 1)); | ||
+ | int i; | ||
+ | for(i=0; i<5; i=i+1) { | ||
+ | dot(origin+1*dir(36+72*i)); | ||
+ | } | ||
+ | |||
+ | label("$A$", origin+1*dir(36+72*0), dir(origin--origin+1*dir(36+72*0))); | ||
+ | label("$B$", origin+1*dir(36+72*1), dir(origin--origin+1*dir(36+72))); | ||
+ | label("$C$", origin+1*dir(36+72*2), dir(origin--origin+1*dir(36+144))); | ||
+ | label("$D$", origin+1*dir(36+72*3), dir(origin--origin+1*dir(36+72*3))); | ||
+ | label("$E$", origin+1*dir(36+72*4), dir(origin--origin+1*dir(36+72*4))); | ||
+ | </asy> | ||
<math> \mathrm{(A) \ 72 } \qquad \mathrm{(B) \ 108 } \qquad \mathrm{(C) \ 120 } \qquad \mathrm{(D) \ 135 } \qquad \mathrm{(E) \ 144 } </math> | <math> \mathrm{(A) \ 72 } \qquad \mathrm{(B) \ 108 } \qquad \mathrm{(C) \ 120 } \qquad \mathrm{(D) \ 135 } \qquad \mathrm{(E) \ 144 } </math> |
Revision as of 19:26, 18 August 2011
Problem
Given regular pentagon , a circle can be drawn that is tangent to at and to at . The number of degrees in minor arc is
Solution
Define major arc DA as , and minor arc DA as . Extending DC and AB to meet at F, we see that . We now have two equations: , and . Solving, and .
See also
1995 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 16 |
Followed by Problem 18 | |
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 |