In the figure, $ABCD$ is a rectangle and $EFGH$ is a parallelogram. Using the measurements given in the figure, what is the length $d$ of the segment that is perpendicular to $HE$ and $FG$? [asy] defaultpen(linewidth(0.8)); size(200); pair A=(0,8), B=(10,8), C=(10,0), D=origin; pair E=(4,8), F=(10,3), G=(6,0), H=(0,5); pair I=H+4*dir(H--E); pair J=foot(I, F, G); draw(A--B--C--D--cycle); draw(E--F--G--H--cycle); draw(I--J); draw(rightanglemark(H,I,J)); draw(rightanglemark(F,J,I)); label("$A$", A, dir((5,4)--A)); label("$B$", B, dir((5,4)--B)); label("$C$", C, dir((5,4)--C)); label("$D$", D, dir((5,4)--D)); label("$E$", E, dir((5,4)--E)); label("$F$", F, dir((5,4)--F)); label("$G$", G, dir((5,4)--G)); label("$H$", H, dir((5,4)--H)); label("$d$", I--J, SW); label("3", H--A, W); label("4", E--A, N); label("6", E--B, N); label("5", F--B, dir(1)); label("3", F--C, dir(1)); label("5", H--D, W); label("4", C--G, S); label("6", D--G, S); [/asy] $ \textbf{(A)}\ 6.8\qquad\textbf{(B)}\ 7.1\qquad\textbf{(C)}\ 7.6\qquad\textbf{(D)}\ 7.8\qquad\textbf{(E)}\ 8.1 $