Difference between revisions of "1954 AHSME Problems/Problem 46"
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| + | == Problem 46== | ||
| + | |||
| + | In the diagram, if points <math>A, B</math> and <math>C</math> are points of tangency, then <math>x</math> equals: | ||
| + | |||
| + | <asy> | ||
| + | unitsize(5cm); | ||
| + | defaultpen(linewidth(.8pt)+fontsize(8pt)); | ||
| + | dotfactor=3; | ||
| + | pair A=(-3*sqrt(3)/32,9/32), B=(3*sqrt(3)/32, 9/32), C=(0,9/16); | ||
| + | pair O=(0,3/8); | ||
| + | draw((-2/3,9/16)--(2/3,9/16)); | ||
| + | draw((-2/3,1/2)--(-sqrt(3)/6,1/2)--(0,0)--(sqrt(3)/6,1/2)--(2/3,1/2)); | ||
| + | draw(Circle(O,3/16)); | ||
| + | draw((-2/3,0)--(2/3,0)); | ||
| + | label("$A$",A,SW); | ||
| + | label("$B$",B,SE); | ||
| + | label("$C$",C,N); | ||
| + | label("$\frac{3}{8}$",O); | ||
| + | draw(O+.07*dir(60)--O+3/16*dir(60),EndArrow(3)); | ||
| + | draw(O+.07*dir(240)--O+3/16*dir(240),EndArrow(3)); | ||
| + | label("$\frac{1}{2}$",(.5,.25)); | ||
| + | draw((.5,.33)--(.5,.5),EndArrow(3)); | ||
| + | draw((.5,.17)--(.5,0),EndArrow(3)); | ||
| + | label("$x$",midpoint((.5,.5)--(.5,9/16))); | ||
| + | draw((.5,5/8)--(.5,9/16),EndArrow(3)); | ||
| + | label("$60^{\circ}$",(0.01,0.12)); | ||
| + | dot(A); | ||
| + | dot(B); | ||
| + | dot(C);</asy> | ||
| + | |||
| + | <math> \textbf{(A)}\ \frac{3}{16}"\qquad\textbf{(B)}\ \frac{1}{8}"\qquad\textbf{(C)}\ \frac{1}{32}"\qquad\textbf{(D)}\ \frac{3}{32}"\qquad\textbf{(E)}\ \frac{1}{16}" </math> | ||
==Solution 1== | ==Solution 1== | ||
Revision as of 16:01, 27 April 2020
Problem 46
In the diagram, if points 
and 
are points of tangency, then 
equals:
![[asy] unitsize(5cm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=3; pair A=(-3*sqrt(3)/32,9/32), B=(3*sqrt(3)/32, 9/32), C=(0,9/16); pair O=(0,3/8); draw((-2/3,9/16)--(2/3,9/16)); draw((-2/3,1/2)--(-sqrt(3)/6,1/2)--(0,0)--(sqrt(3)/6,1/2)--(2/3,1/2)); draw(Circle(O,3/16)); draw((-2/3,0)--(2/3,0)); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,N); label("$\frac{3}{8}$",O); draw(O+.07*dir(60)--O+3/16*dir(60),EndArrow(3)); draw(O+.07*dir(240)--O+3/16*dir(240),EndArrow(3)); label("$\frac{1}{2}$",(.5,.25)); draw((.5,.33)--(.5,.5),EndArrow(3)); draw((.5,.17)--(.5,0),EndArrow(3)); label("$x$",midpoint((.5,.5)--(.5,9/16))); draw((.5,5/8)--(.5,9/16),EndArrow(3)); label("$60^{\circ}$",(0.01,0.12)); dot(A); dot(B); dot(C);[/asy]](http://latex.artofproblemsolving.com/4/0/3/403cbf1c989b92c4861af94452e599258e01adf4.png)
