Difference between revisions of "1995 AHSME Problems/Problem 17"

m (Problem: <asy> by dragon96)
(Fixed mislabeling in the diagram.)
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}
 
}
  
label("$A$", origin+1*dir(36+72*0), dir(origin--origin+1*dir(36+72*0)));
+
label("$B$", 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("$A$", 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("$E$", 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("$D$", 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)));
+
label("$C$", origin+1*dir(36+72*4), dir(origin--origin+1*dir(36+72*4)));
 
</asy>
 
</asy>
  

Revision as of 23:54, 18 August 2011

Problem

Given regular pentagon $ABCDE$, a circle can be drawn that is tangent to $\overline{DC}$ at $D$ and to $\overline{AB}$ at $A$. The number of degrees in minor arc $AD$ is

[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("$B$", origin+1*dir(36+72*0), dir(origin--origin+1*dir(36+72*0))); label("$A$", origin+1*dir(36+72*1), dir(origin--origin+1*dir(36+72))); label("$E$", 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("$C$", origin+1*dir(36+72*4), dir(origin--origin+1*dir(36+72*4))); [/asy]

$\mathrm{(A) \ 72 } \qquad \mathrm{(B) \ 108 } \qquad \mathrm{(C) \ 120 } \qquad \mathrm{(D) \ 135 } \qquad \mathrm{(E) \ 144 }$

Solution

Define major arc DA as $DA$, and minor arc DA as $da$. Extending DC and AB to meet at F, we see that $\angle CFB=36=\frac{DA-da}{2}$. We now have two equations: $DA-da=72$, and $DA+da=360$. Solving, $DA=216$ and $da=144\Rightarrow \mathrm{(E)}$.

See also

1995 AHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 16
Followed by
Problem 18
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