(i) Measurement of AD.
We can see that AD is a side in a right triangle with the other sides 41 and 40. Hypotenuse is not AD. Use Pythagorean Theorem.
AD=\sqrt{41^2-40^2} \\ AD=\sqrt{1681-1600} \\ AD=\sqrt{8} \\ AD=9

(ii)Area of Quadrilateral ABCD.
We can see AD is the height. The bases are 15 and 40. Follow formula:
A=\frac{a+b}2h \\ A=\frac9{15+40}2 \\ A=9\times27.5 \\ A=247.5

Ans: AD=9cm       A=247.5cm²