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Which right triangles have area equal to perimeter?

+2 votes
Which right triangles have integer sides such that the area and perimeter are equal?
asked Jul 10, 2014 in Math by emily-gan Evoker (770 points)  

1 Answer

+1 vote
(a,b,c)= (5,12,13), (6,8,10), (6,25,29), (7,15,20), (9,10,17)
answered Jul 10, 2014 by luongamanda Evoker (610 points)  

So the triangle side lengths (a, b, c) for the first one are (5, 12, 13), respectively.

The perimeter is then P = a + b + c = 5 + 12 + 13 = 30 and the semiperimeter, call it s, is therefore 15.

Using Heron's formula to find the area, \( A = \sqrt{s \cdot (s-a)(s-b)(s-c)} \), we get

\( A = \sqrt{15 \cdot (15-5)(15-12)(15-13)} \)

\( A = \sqrt{15 \cdot (10)(3)(2)} \)

\( A = \sqrt{900} \)

\( A = 30 \)

Nice. I wonder what is the general approach to finding these? 

Maybe if we start with P = A, or \(a + b + c = \sqrt{s \cdot (s-a)(s-b)(s-c)}\).

But this is one equation with three unknowns surprise

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