AMC 12 2003b Problem 22

Let $ABCD$ be a rhombus with $AC = 16$ and $BD = 30$. Let $N$ be a point on $\overline{AB}$, and let $P$ and $Q$ be the feet of the perpendiculars from $N$ to $\overline{AC}$ and $\overline{BD}$, respectively. Which of the following is closest to the minimum possible value of $PQ$?

size(200);
defaultpen(0.6);
pair O = (15*15/17,8*15/17), C = (17,0), D = (0,0), P = (25.6,19.2), Q = (25.6, 18.5);
pair A = 2*O-C, B = 2*O-D;
pair P = (A+O)/2, Q=(B+O)/2, N=(A+B)/2;
draw(A--B--C--D--cycle);
draw(A--O--B--O--C--O--D);
draw(P--N--Q);
label("$A$",A,WNW);
label("$B$",B,ESE);
label("$C$",C,ESE);
label("$D$",D,SW);
label("$P$",P,SSW);
label("$Q$",Q,SSE);
label("$N$",N,NNE);

$\mathrm{(A)}\ 6.5 \mathrm{(B)}\ 6.75 \mathrm{(C)}\ 7 \mathrm{(D)}\ 7.25 \mathrm{(E)}\ 7.5$