Try this beautiful Problem on Geometry based on External Tangent from AMC 10 A, 2018. You may use sequential hints to solve the problem.
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 $A B$ can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n ?$
,
Geometry
Triangle
Pythagoras
Pre College Mathematics
AMC-10A, 2018 Problem-15
$69$
Given that 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 $B$. we have to find out the length \(AB\).
Now join \(A\) & \(B\) and the points \(Y\) & \(Z\). If we can show that \(\triangle XYZ \sim \triangle XAB\) then we can find out the length of \(AB\).
Now can you finish the problem?
now the length of \(YZ=5+5=10\) (as the length of the radius of smaller circle is $5$) and \(XY=XA-AY=13-5=8\). Now \(YZ|| AB\).therefore we can say that \(\triangle XYZ \sim \triangle XAB\). therefore we can write $\frac{X Y}{X A}=\frac{Y Z}{A B}$
Now Can you finish the Problem?
From the relation we can say that $\frac{X Y}{X A}=\frac{Y Z}{A B}$
\(\Rightarrow \frac{8}{13}=\frac{10}{AB}\)
\(\Rightarrow AB=\frac{13\times 10}{8}\)
\(\Rightarrow AB=\frac{65}{4}\) which is equal to \(\frac{m}{n}\)
Therefore \(m+n=65+4=69\)
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