I'm assuming you're familiar with Law of Sines/Cosines.

To write a FULL step by step solution here would be almost impossible;

classroom environment required (blackboard, chalk.....).

I'll give you a "how to get started".

Draw an approximate representative diagram (quad ABCD); join AC.

Let AB = b, AC = c, AD = d, average speed = x.

Get b,c,d in terms of x (assuming you're familiar with speed = distance/time):

b = (7x - 120) / 12, c = x / 2, d = (2x - 30) / 3

(notice that x > 120/7)

Now you need to calculate a few angles, like B; using triangle ABC:

B = arccos[(b^2 + 100 - c^2) / (20b)] : the 100 is 10^2 (side BC).

It's sure not an easy problem! Good luck.