MHB Triangles: Short Leg-Longer Leg-Hypotenuse a=448

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The discussion revolves around determining the lengths of the sides of four right triangles, given the short leg (a = 448) and the relationships among the sides. The initial attempt suggested non-integer values for some sides, but it was clarified that all lengths must be integers. The lowest integer solution found was (448, 975, 1073, 495, 952, 840), with another solution being (840, 3875, 3965, 1364, 3723, 3627). The participants noted the surprising scarcity of valid integer combinations for these triangles.
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4 right triangles; short leg - longer leg - hypotenuse :
a - b - c
d - e - c
d - f - b
a - f - e

HINT: a = 448 (which to my surprise is lowest case)
 
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Wilmer said:
4 right triangles; short leg - longer leg - hypotenuse :
a - b - c
d - e - c
d - f - b
a - f - e

HINT: a = 448 (which to my surprise is lowest case)

Hi Wilmer, :)

So I think the challenge here is to find the lengths of \(a,b,c,d,e\mbox{ and }f\). Isn't? I don't know if there's a unique answer to this question. But it seems to me that,

\[a=3,\,b=4,\,c=5,\,d=1,\,e=\sqrt{24},\,f=\sqrt{15}\]

satisfies all the restrictions that you have imposed.

Kind Regards,
Sudharaka.
 
Sorry...all lengths are integers.
Forgot to mention that.

Lowest I could find:
(a, b, c, d, e, f) = (448, 975, 1073, 495, 952, 840)

And next one is: (840,3875,3965,1364,3723,3627).

I was surprised at how few there are...
 
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