Difference between revisions of "1981 AHSME Problems/Problem 27"
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In the adjoining figure triangle <math>ABC</math> is inscribed in a circle. Point <math>D</math> lies on <math>\stackrel{\frown}{AC}</math> with <math>\stackrel{\frown}{DC} = 30^\circ</math>, and point <math>G</math> lies on <math>\stackrel{\frown}{BA}</math> with <math>\stackrel{\frown}{BG}\, > \, \stackrel{\frown}{GA}</math>. Side <math>AB</math> and side <math>AC</math> each have length equal to the length of chord <math>DG</math>, and <math>\angle CAB = 30^\circ</math>. Chord <math>DG</math> intersects sides <math>AC</math> and <math>AB</math> at <math>E</math> and <math>F</math>, respectively. The ratio of the area of <math>\triangle AFE</math> to the area of <math>\triangle ABC</math> is | In the adjoining figure triangle <math>ABC</math> is inscribed in a circle. Point <math>D</math> lies on <math>\stackrel{\frown}{AC}</math> with <math>\stackrel{\frown}{DC} = 30^\circ</math>, and point <math>G</math> lies on <math>\stackrel{\frown}{BA}</math> with <math>\stackrel{\frown}{BG}\, > \, \stackrel{\frown}{GA}</math>. Side <math>AB</math> and side <math>AC</math> each have length equal to the length of chord <math>DG</math>, and <math>\angle CAB = 30^\circ</math>. Chord <math>DG</math> intersects sides <math>AC</math> and <math>AB</math> at <math>E</math> and <math>F</math>, respectively. The ratio of the area of <math>\triangle AFE</math> to the area of <math>\triangle ABC</math> is | ||
| − | + | <asy> | |
| + | defaultpen(linewidth(.8pt)); pair C = origin; pair A = 2.5*dir(75); pair B = A + 2.5*dir(-75); path circ =circumcircle(A,B,C); pair D = waypoint(circ,(7/12)); pair G = waypoint(circ,(1/6)); pair E = intersectionpoint(D--G,A--C); pair F = intersectionpoint(A--B,D--G); label("$A$",A,N); label("$B$",B,SE); label("$C$",C,SW); label("$D$",D,SW); label("$G$",G,NE); label("$E$",E,NW); label("$F$",F,W); label("$30^\circ$",A,12S+E,fontsize(6pt)); draw(A--B--C--cycle); draw(circ); draw(Arc(A,0.25,-75,-105)); draw(D--G); | ||
| + | </asy> | ||
| + | |||
<math>\textbf{(A)}\ \dfrac {2 - \sqrt {3}}{3}\qquad \textbf{(B)}\ \dfrac {2\sqrt {3} - 3}{3}\qquad \textbf{(C)}\ 7\sqrt {3}-12\qquad \textbf{(D)}\ 3\sqrt {3}-5\qquad\\ \textbf{(E)}\ \dfrac {9-5\sqrt {3}}{3}</math> | <math>\textbf{(A)}\ \dfrac {2 - \sqrt {3}}{3}\qquad \textbf{(B)}\ \dfrac {2\sqrt {3} - 3}{3}\qquad \textbf{(C)}\ 7\sqrt {3}-12\qquad \textbf{(D)}\ 3\sqrt {3}-5\qquad\\ \textbf{(E)}\ \dfrac {9-5\sqrt {3}}{3}</math> | ||
==Solution== | ==Solution== | ||
Latest revision as of 10:11, 1 December 2022
Problem
In the adjoining figure triangle 
is inscribed in a circle. Point 
lies on 
with 
, and point 
lies on 
with 
. Side 
and side 
each have length equal to the length of chord 
, and 
. Chord intersects sides and at 
and 
, respectively. The ratio of the area of 
to the area of 
is
![[asy] defaultpen(linewidth(.8pt)); pair C = origin; pair A = 2.5*dir(75); pair B = A + 2.5*dir(-75); path circ =circumcircle(A,B,C); pair D = waypoint(circ,(7/12)); pair G = waypoint(circ,(1/6)); pair E = intersectionpoint(D--G,A--C); pair F = intersectionpoint(A--B,D--G); label("$A$",A,N); label("$B$",B,SE); label("$C$",C,SW); label("$D$",D,SW); label("$G$",G,NE); label("$E$",E,NW); label("$F$",F,W); label("$30^\circ$",A,12S+E,fontsize(6pt)); draw(A--B--C--cycle); draw(circ); draw(Arc(A,0.25,-75,-105)); draw(D--G); [/asy]](http://latex.artofproblemsolving.com/2/b/8/2b826c3c9e9b8ac944aa21f15bb5d94f14e1e54c.png)
