let

'a' and 'b' be the legs of the triangle and we will assume without losing generality a>b that is b is the shorter leg

'c' the hypotenuse of the triangle

the hypotenuse of a right triangle is 20 inches longer than the shorter leg

c = b + 20

the longer leg is 1 inches longer than the shorter leg

a = b + 1

using the Pythagorean Theorem we can write

a^2 + b^2 = c^2

(b+1)^2 + b^2 = (b + 20)^2

by solving the quadratic equation we find

b = 19 + 2*(190^0.5)

b = 46.5681 in approximately

we consider just the positive roots of the quadratic equation since the length must be >0

The length of the shorter leg is 46.5681 in approximately.