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A problem I'm working on

Find all right angle triangles with integer sides with perimeter 60 units.

I'm working on this problem, any ideas?

Note by Swapnil Das
1 year, 10 months ago

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We will try and get a Diophantine equation you can factor. Let \(x\) and \(y\) be the 2 shorter sides of the right angle triangle.






We also have \(x, y < 60\) since the perimeter must be 60. So, both brackets must be negative. Also, \(x, y > 0\) since our triangle is non-degenerate. This implies that \(-60<x-60<0, -60<y-60<0\). We can use this to remove certain factors of 1800.

Now we list all factors of 1800 which its complement that also satisfy the above ranges.

\[1800 = (-36, -50), (-40, -45)\]

From this, we get \((x, y)\) values of \((24, 10)\) and \((20, 15)\). Thus, the only triangles which satisfy are the 10-24-26 triangle and the 15-20-25 triangle.

Note: I didn't bother with solutions \((10, 24)\) and \((15, 20)\) since they gave the same triangles.

Sharky Kesa - 1 year, 10 months ago

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