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?

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Comments

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.

Sharky Kesa - 5 years, 5 months 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}`	$\boxed{123}$