Puzzle: Right triangle area and perimeter
February 22, 2009 am28 4:42 am
In the same spirit as the previous puzzle (which generated some nice discussion/solutions), let’s keep to the same theme:
Find all the right triangles for which the area equals the perimeter. (just the numbers. ignore the mismatched units)
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JD2718 
From some pretty torturous algebra I got any legs x and y for which y = (4x-8)/(x-4) which helpfully yields (5,12) and (6,8) (and the reverse) which I think are the only integral solutions. Domain is 0<x 4.
I agree with Kate (apart from the domain, which either confuses me or is not what she meant), but again I’d phrase it differently: let the legs be (x+4) and (y+4). Then the solutions are exactly those positive x,y for which xy=8.
(This is less easy to show, and figuring out why they must be positive took me a couple of minutes too :).)