# 1967 AHSME Problems/Problem 15

## Problem

The difference in the areas of two similar triangles is $18$ square feet, and the ratio of the larger area to the smaller is the square of an integer. The area of the smaller triange, in square feet, is an integer, and one of its sides is $3$ feet. The corresponding side of the larger triangle, in feet, is: $\textbf{(A)}\ 12\quad \textbf{(B)}\ 9\qquad \textbf{(C)}\ 6\sqrt{2}\qquad \textbf{(D)}\ 6\qquad \textbf{(E)}\ 3\sqrt{2}$

## Solution

If the areas of the two triangles are $A_1$ and $A_2$ with $A_1 > A_2$, we are given that $A_1 - A_2 = 18$ and $\frac{A_1}{A_2} = k^2$

Plugging in the second equation into the first leads to $(k^2 - 1)A_2 = 18$. If $A_2$ is an integer, it can only be a factor of $18$ - namely $1, 2, 3, 6, 9, 18$. Since $A_1 = A_2 + 18$, this would lead to $A_1 = 19, 20, 21, 24, 27, 36$. This would lead to $\frac{A_1}{A_2} = 19, 10, 7, 4, \frac{8}{3}, \frac{3}{2}$, and the only square is $4$.

Thus, the ratio of the areas is $4$, and hence the ratio of the sides is $\sqrt{4} = 2$. The corresponding side has a length of $3 \cdot 2 = 6$, which is option $\fbox{D}$.

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