Make a circle with the outer center of triangle ABC as O point.

O is OE, BC is perpendicular to E, so that OP is perpendicular to AD in P.

Because angle BAC=30 degrees, angle BOC=2 times angle BAC=60 degrees.

Because OB=OC? So the triangle OBC is an equilateral triangle.

So BC= 10, EB=EC=5, OE=5 times the root number 3.

So DP is also equal to 5 times the root number 3 (because DPOE is rectangular).

OP=DE=5-4= 1

OA is also a radius, which is equal to 10.

So in the right triangle AOP

According to Pythagorean theorem, the root sign of PA=3 times 1 1 is obtained.

So ad = AP+PD = triple root number 1 1+5 root number 3.

My picture is a little exaggerated, so I can make do with it.