Difference between revisions of "1954 AHSME Problems/Problem 46"

Line 1: Line 1:
 +
== 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 17:01, 27 April 2020

Problem 46

In the diagram, if points $A, B$ and $C$ are points of tangency, then $x$ 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]

$\textbf{(A)}\ \frac{3}{16}"\qquad\textbf{(B)}\ \frac{1}{8}"\qquad\textbf{(C)}\ \frac{1}{32}"\qquad\textbf{(D)}\ \frac{3}{32}"\qquad\textbf{(E)}\ \frac{1}{16}"$

Solution 1

Invalid username
Login to AoPS