The formula for area of a triangle is A=b*h/2, where b and h base and height of a triangle. In the case of a right triangle, they are the legs. So we have two unknowns and 2 equations.
b^2 + h^2 = 144 (Pythagorean theorem)
Solve the first equation for either b or h and substitute into the second.
48=b*h (multiply both sides by 2)
b=48/h (divide both sides by h)
(48/h)^2 + h^2 = 144 (substitute 48/h for b)
2304/h^2 + h^2 = 144
2304/h^2 + h^4/h^2 = 144 (find common denominator for left side)
2304 + h^4 = 144*h^2 (multiply both sides by h^2)
This is really a quadratic equation. You can make this clearer by letting x = h^2
2304 + x^2 = 144x
x^2 - 144x + 2304 = 0
Use the quadratic formula to solve.
x = (144 ± sqrt((-144)^2-4*1*2304)))/2 ≈ 125.665, 18.334
If x = h^2 then sqrt(x) = h
h ≈ 11.210 or 4.282
Then plug into b = 48/h to get b. You'll find that if you choose 11.210 for h you'll bet 4.282 for b and if you choose 4.282 for h you'll get 11.210 for b.
So the legs of the triangle are approximately 4.282 and 11.210 inches.
Thanks for the interesting problem.