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

I'm working on this problem, any ideas?

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

I'm working on this problem, any ideas?

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

\[x+y+\sqrt{x^2+y^2}=60\]

\[60-x-y=\sqrt{x^2+y^2}\]

\[3600+x^2+y^2-120x-120y+2xy=x^2+y^2\]

\[xy-60x-60y+1800=0\]

\[(x-60)(y-60)=1800\]

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 ago

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