There are two triangles here: ABC and ADF. These triangles are similar, and thus proportional. So, we can set up a proportion between the left side of each triangle and the base of each triangle.
AB / AD = BC / DE
4 / (4 + BD) = 5 / 9
---AD is made up of AB and DB. We know AB, but not BD, so we need to find its value to find the length of AD.
5(4 + BD) = 9 x 4
20 + 5BD = 36
5BD = 16
BD = 3.2
AD = AB + BD
AD = 4 + 3.2
AD = 7.2
Hope this helps!