In a right triangle,if the measure of one of the angles is $30^{\circ}$,then the side opposite to the angle with measure $30^{\circ}$ is .......... times the hypotenuse.

From a point $h$ metres above the water level of a lake,the angle of elevation of the top of a palace is found to be $\alpha$ and the angle of depression of the image of the top of the palace observed in the lake is found to be $\beta$. Find the height of the palace.

From the top of a tower $h \, m$ high,the angles of depression of two objects,which are in line with the foot of the tower,are $\alpha$ and $\beta$ (where $\beta > \alpha$). Find the distance between the two objects.

The height of a building is $11 \, m$ and the height of a pole is $8 \, m$. Watching from the midpoint of the line segment joining their bases,the angle of elevation of the top of the pole is $\alpha$ and that of the top of the building is $\beta$. Then,$\ldots \ldots \ldots$ holds true.

$A$ vertical tower stands on a horizontal plane and is surmounted by a vertical flag staff of height $h$. At a point on the plane,the angles of elevation of the bottom and the top of the flag staff are $\alpha$ and $\beta$,respectively. Prove that the height of the tower is $\left(\frac{h \tan \alpha}{\tan \beta-\tan \alpha}\right)$.

On walking for search of a ball $x$ metres on a hill making an angle of $30^{\circ}$ with the ground,one can reach a height of $y$ metres from the ground. Then $\ldots \ldots \ldots . . .$