I was looking at this question the other day and thinking of an easy way to solve it....then I saw CPHIL's answer which was different than my thinking, but I wil use his graph diagram rather than uploading my hand drawings.....
The short thick black line is = 3 (given in the question) the slope of the blue line is - 12/5 (given).... Now let's rotate the triangle so the blue line is verticle ..like this:
Now the thin black line (hypotenuse, h) that we are looking for has a slope of 12/5
We can find the length of the blue line, b, by TAN = opp/adj = 12/5 = b/3 so b = 36/5
Now, use Pythag theorom to find the hypotenuse (the distance on the y axis betweeen the two parallel lines)
h^2= (36/5)^2 + 3^2 results in h= 7.8
we need TWO h's to find the distance between the TWO lines parallel to the given line: 2 x 7.8 = 15.6
Hope this isn't too confusing !