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I came across the following problem and I know for certain the answer is $6$, but am unsure as to where this comes from.

Let $m$ be the sum of the digits in 4950. The difference in the area of two similar triangles is $m$ square feet, and the ratio of the larger area to the smaller area is the square of an integer. The area of the smaller triangle, in square feet, is an integer and one of its sides is $3$ feet. Find the length of the corresponding side of the larger triangle in feet.

Besides the obvious fact that $m=18$. I am unsure as to how to approach this problem.

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    $\begingroup$ I think it means "$m = 4 + 9 + 5 + 0$." $\endgroup$ – John Hughes Apr 12 '17 at 0:49
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Let $r$ be the ratio of the two triangles and let $a$ be the area of the smaller one. Then the area of the bigger one is given by $a+18$ and $a\cdot r^2$. Hence, $$ ar^2=a+18 \Longleftrightarrow a(r^2-1)=18$$ So 18 is divisible by $(r^2-1) $ and therefore $r^2-1 <18$ which implies $r^2-1=0$, $r^2-1=3$, $r^2-1=8$ or $r^2-1=15$ since $r \in \mathbb {Z}$. Of these four possible values for $r^2-1$, only $3$ divides $18$. Therefore, $r^2-1=3$ and $r=2$. So the length of the side in the bigger triangle that corresponds to the side of length $3$ in the smaller one has length $6$.

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"Let $m$ be the sum of the digits in 4950" $\Rightarrow m=18$

Let $k$ be coefficient of similarity then

"The difference in the area of two similar triangles is m square feet" $\Rightarrow (k^2-1)S=18$

"the ratio of the larger area to the smaller area is the square of an integer" $\Rightarrow k\in\mathbb N$

"The area of the smaller triangle, in square feet, is an integer" $\Rightarrow S\in \mathbb N$

Since $(k^2-1)S=18$ and both divisors are integers it follows that $S\in\{1,2,3,6,9,18\}\Rightarrow k=2$

So length of the corresponding (to those of length 3 feet) side of the larger triangle is $6$ feet.

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