The height of the parallelogram, drawn from the vertex of the obtuse angle, is equal to $a$ and bisects the side. The acute angle of the parallelogram is $30^{\circ}$.

Find the diagonals of the parallelogram.
Let $M$ be the foot of the altitude $B M$ of parallelogram $A B C D$, drawn from the vertex of the obtuse angle $B$. By condition, $M$ is the midpoint of $A D$. Then

$\angle B D M=\angle B A M=30^{\circ}$. From the right triangle $B M D$ we find that $B D=2 B M=2 a$.

Let $K$ be the foot of the perpendicular dropped from vertex $C$ to line $A D$. Then $C K=B M=a, D K=$ $A M=a \sqrt{3}, A K=3 a \sqrt{3}$.

By the Pythagorean theorem $A C^{2}=A K^{2}+C K^{2}=28 a^{2}$.

## Answer

## $2 a, 2 a \sqrt{3}$.