To find the area of the trapezoid, we first need to find the height.
Using the Pythagorean theorem in the right triangle formed by the diagonal and the dotted line, we have:
5^2 = x^2 + 9^2
25 = x^2 + 81
x^2 = 25 - 81
x^2 = 56
x = √56
x ≈ 7.48 cm
Now, we can use the formula for the area of a trapezoid:
Area = (1/2) * (a + b) * h
Area = (1/2) * (4 + 6) * 7.48
Area = (1/2) * 10 * 7.48
Area = 37.4 square centimeters
Therefore, the area of the quadrilateral is 37.4 square centimeters.