1964 AHSME Problems/Problem 13

Problem 13

A circle is inscribed in a triangle with side lengths $8, 13$ , and $17$ . Let the segments of the side of length $8$ , made by a point of tangency, be $r$ and $s$ , with $r<s$ . What is the ratio $r:s$ ?

$\textbf{(A)}\ 1:3 \qquad \textbf{(B)}\ 2:5 \qquad \textbf{(C)}\ 1:2 \qquad \textbf{(D)}\ 2:3 \qquad \textbf{(E)}\ 3:4$

Solution

Label our triangle $ABC$ where that $AB=17$ , $AC=13$ , and $BC=8$ . Let $L$ , $J$ , and $K$ be the tangency points of $BC$ , $AC$ , and $AB$ respectively. Let $AK=x$ , which implies $AJ=x$ . Thus $KB=BL=17-x$ and $JC=LC=13-x$ .

Since $BL+LC=(17-x)+(13-x)=8$ , $x=11$ . Thus $r:s=CL:BL=13-x:17-x=2:6=1:3$ , hence our answer is $\fbox{A}$ .

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1964 AHSME Problems/Problem 13

Problem 13

Solution