1996 AMC 12 Problem 19

Compare the distance between adjacent side midpoints with the original side length

The smaller and larger hexagons are similar, so square their side-length ratio

Two adjacent midpoints and their shared vertex form a triangle with two sides $\frac{s}{2}$ and included angle $120^\circ.$ Its opposite side has squared length $$\begin{aligned} \frac{s^2}{4}+\frac{s^2}{4} &-2\left(\frac{s}{2}\right)^2\cos120^\circ\\ &=\frac{3s^2}{4}. \end{aligned}$$ Thus the smaller hexagon has scale factor $\frac{\sqrt3}{2},$ and its area ratio is $\frac{3}{4}.$ The correct answer is D.