Rhombus with integer side lengths

Hello, I would like to discuss the following question:

A rhombus has an area of 120. Given that the diagonals and sides are all integer values, what is the length of the side of the rhombus?

Here is one way of solving the problem: I listed all the possible pairs that multiply to 240 (since Area = pq/2) and used the Pythagorean theorem to find out whether there are any integer side lengths. After a while I found that 10 and 12 are the diagonals which result in a Pythagorean triple with integer side lengths of 5,12 and 13. Hence 13 is the answer.

Are there any other possible ways to solve this problem? Please let me know if you have any ideas or approaches!

#Geometry

Easy Math Editor

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Comments

Here's a bit of an update:

I then plotted y=\sqrt((x^(4)+3600)/(x^(2)))) on a graphing software and unsurprisingly, the points (5,13) and (12,13) both lie on the graph! (as well as (-5,13) and (-12,13) )

I used desmos.com (an amazing graphing software by the way)

This doesn't answer the question completely but that's all I have for now.

Ethan Mandelez - 6 months, 3 weeks ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$