From the figure, it shows that the quadrilateral is made up with two right triangles with their sides are exactly Pythagorean triplets. Let's confirm our hypothesis.

First, determine the length of hypotenuse of \triangle ABD

Since the length of two legs \triangle ABD are 3 and 4 respectively, we know the length of its hypotenuse can be computed by using Pythagorean Theorem.

BD^2 = AD^2+AB^2

Then,

BD = \sqrt{3^2+4^2} =5

Now for \triangle BCD, the measure of three sides meet the Pythagorean Theorem

13^2 = 12^2+5^2 =169

implies

BC^2 = BD^2+CD^2

Therefore, \triangle BCD is also a right triangle.

So the area of quadrilateral ABCD is the sum of the two right triangle.

A_{ABCD} =A_{\triangle ABD }+A_{\triangle BCD }

=\dfrac{1}{2}\cdotp 3\cdotp 4+\dfrac{1}{2}\cdotp 5\cdotp 12

=36

It turns out that the area of quadrilateral ABCD 36. E is the correct choice.