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I've tried many methods of solving this including tables and the inradius of a rhombus formula, but I can't get far on this one. Does anyone see an efficient way to go about this problem?

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    Is this really supposed to be done by hand? – 2017-02-27
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    You can use any aid you have (calculator, computer) – 2017-02-27
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    Then it's easy -- the answer is 240/17 – 2017-02-27
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    How on earth did you get that? Your answer lines up with the official key too, I'm sure you already knew this though lol. – 2017-02-27
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    Not a complete brute force search, but at some point, I tested the remaining possibilities with a program. – 2017-02-27
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    Where does the problem come from? – 2017-02-27
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    This question is from a high school level math competition state final, and some (a very select few) kids do actually bring in computers, so I suppose this question was designed to keep even those kids from getting perfect scores while still making it possible. Only 1% of anyone got this question correct, so I guess we'll just call this one a computer problem lol! – 2017-02-27
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    Was it a multiple choice problem? – 2017-02-27
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    Nope, write-in answers only – 2017-02-27
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    It's doable by hand, but awful. – 2017-02-27
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    Yeah these tests are 20 questions 50 minutes so this problem is likely just there to suck in those who are bad at managing time and will spend 20 minutes on this single question. And also maybe to give people who bring in computers a challenge. – 2017-02-27
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    Let $d,e$ be the diagonal lengths, with $d \le e$. One can show the following without any computational aids: \begin{align*} &\bullet\;\;d^2 + e^2 \text{ must be a perfect square}\\ &\bullet\;\;29\le d\le 39\\ &\bullet\;\;43\le e \le 70 \end{align*} – 2017-02-27
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    But if it's just one question in a 20 question, 50 minute test, it's a time trap, no matter who tries it. Of course, maybe there's a simple way that I missed. – 2017-02-27
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    Yeah, maybe someone else will come across something, thanks for your assistance, though! – 2017-02-27

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Here's a semi-brute-force version using a program ...

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    Estimate: 2 minutes of math; 5 minutes of programming; 1 second to execute. – 2017-02-27
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    I'm going to go ahead and see if anyone else can come up with any clever methods, maybe using tables, if not I'll just accept yours and move on haha – 2017-02-27