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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
11 months, 3 weeks 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.

$x+y+\sqrt{x^2+y^2}=60$

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

$3600+x^2+y^2-120x-120y+2xy=x^2+y^2$

$xy-60x-60y+1800=0$

$(x-60)(y-60)=1800$

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. · 11 months, 3 weeks ago