+128474 Let x be the side of the triangle
Using the Pythagorean Theorem we have this relationship
sqrt ( x^2 - 9) + sqrt (x^2 - 16) = sqrt (x^2 -1) square both sides
x^2 -9 + 2 sqrt [ (x^2 - 9)(x^2 -16) ] + x^2 -16 = x^2 -1
-25 + 2x^2 + 2sqrt [ x^4 - 25x^2 + 144 ] = x^2 -1
2sqrt [ x^4 - 25x^2 + 144 ] = 24 - x^2 square both sides again
4[ x^4 - 25x^2 + 144] = x^4 - 48x^2 + 576
4x^4 - 100x^2 + 576 = x^4 - 48x^2 + 576 simplify
3x^4 = 52x^2
x^2 ( 3x^2 - 52) = 0
x^2 = 52/3
x = sqrt (52) /sqrt (3) = 2sqrt (13)/ sqrt (3) = (2/3)sqrt (39) ≈ 4.163
