Perpendicular Diagonal

Let the two diagonals intersect at $O$ , $AO=a$ , $BO=b$ , $CO=c$ , and $DO=d$ . Then by Pythagorean theorem we have:

$\begin{cases} a^2 + b^2 = AB^2 = 1^2 = 1 & ...(1) \\ b^2 + c^2 = BC^2 = 10^2 = 100 & ...(2) \\ c^2 + d^2 = CD^2 = 18^2 = 324 & ...(3) \end{cases}$

From $(1)-(2)+(3): \ a^2 + d^2 = 1-100+324 = 225$ . Note that $DA^2 = a^2 + d^2 = 225$ , $\implies DA = \boxed {15}$ .