Difference between revisions of "2018 AMC 10A Problems/Problem 15"

(Created page with "Two circles of radius 5 are externally tangent to each other and are internally tangent to a circle of radius 13 at points <math>A</math> and <math>B</math>, as shown in the...")
 
Line 10: Line 10:
  
 
<math>\textbf{(A) }  21  \qquad    \textbf{(B) }  29  \qquad    \textbf{(C) }  58  \qquad  \textbf{(D) } 69 \qquad  \textbf{(E) }  93 </math>
 
<math>\textbf{(A) }  21  \qquad    \textbf{(B) }  29  \qquad    \textbf{(C) }  58  \qquad  \textbf{(D) } 69 \qquad  \textbf{(E) }  93 </math>
 +
 +
==Solution==
 +
 +
Call the largest circle <math>\circle X</math>.

Revision as of 15:47, 8 February 2018

Two circles of radius 5 are externally tangent to each other and are internally tangent to a circle of radius 13 at points $A$ and $B$, as shown in the diagram. The distance $AB$ can be written in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?

[asy] draw(circle((0,0),13)); draw(circle((5,-6.2),5)); draw(circle((-5,-6.2),5)); label("$B$", (9.5,-9.5), S); label("$A$", (-9.5,-9.5), S); [/asy]

$\textbf{(A) }   21   \qquad    \textbf{(B) }  29   \qquad    \textbf{(C) }  58   \qquad   \textbf{(D) } 69 \qquad  \textbf{(E) }   93$

Solution

Call the largest circle $\circle X$ (Error compiling LaTeX. Unknown error_msg).