Call the legs of the triangle a and b and the hypotenuse c

let x be the angle of the right triangle such that sin(x) = a / c

We're being asked to look at the maximum value of (a+b) / c (the ratio of the sum of the legs to the hypotenuse)

(a+b)/c = a/c + b/c = sin(x) + cos(x)

This is now a function solely of x, so let's find its extrema:

d/dx (sin(x) + cos(x)) = cos(x) - sin(x) = 0

cos(x) = sin(x) => tan(x) = 1 => x = pi/4 (we can ignore solutions where x > pi/2 since they don't form meaningful right triangles.)

cos(pi/4) = sin(pi/4) = sqrt(2) / 2, so 

sin(pi/4) + cos(pi/4) = sqrt(2) is the extremum

Is this a maximum or a minimum? Well, as x goes arbitrarily close to zero, sin(x) -> 0 and cos(x) -> 1 so sin(x) + cos(x) -> 1 which is less than sqrt(2), and so sqrt(2) is a maximum.