Diagonals of a trapezium $PQRS$ intersect each other at the point $O$. Given $PQ \parallel RS$ and $PQ = 3 RS$. Find the ratio of the areas of triangles $POQ$ and $ROS$.

In $\Delta ABC$ and $\Delta DEF$,$\angle A \cong \angle E$ and $m \angle A + m \angle B = m \angle D + m \angle E$. Then the correspondence $ABC \leftrightarrow \ldots$ between them is a similarity.

$A$ $2.6 \,m$ long ladder leans on a wall such that its lower end remains $1 \,m$ away from the base of the wall. Then,its upper end reaches to the height of .....$m$ on the wall.

For going to a city $B$ from city $A$,there is a route via city $C$ such that $AC \perp CB$,$AC = 2x \, km$ and $CB = 2(x+7) \, km$. It is proposed to construct a $26 \, km$ highway which directly connects the two cities $A$ and $B$. Find how much distance will be saved in reaching city $B$ from city $A$ after the construction of the highway (in $km$).

The correspondence $PQR \leftrightarrow ZYX$ between $\Delta PQR$ and $\Delta XYZ$ is a similarity. If $PQ : ZY = 5 : 3$ and $PR = 10$,find $ZX$.

In rhombus $ABCD$,$\overline{AC} \cap \overline{BD} = \{O\}$. Then the area of $\Delta OAB = \ldots \ldots \ldots \ldots \times$ area of $\square ABCD$.