In trapezium $ABCD$,$AB || CD$. Points $P$ and $Q$ are the midpoints of $AD$ and $BC$ respectively. If $AB = 18 \, cm$ and $PQ = 15 \, cm$,then $CD = \dots \, cm$.

In $\Delta ABC$,$P$,$Q$ and $R$ are the midpoints of $AB$,$BC$ and $CA$ respectively,then $\angle BAC = \angle \ldots \ldots \ldots$

$D, E$ and $F$ are respectively the mid-points of the sides $BC, CA$ and $AB$ of a $\Delta ABC$. Show that $\operatorname{ar}( BDEF ) = \frac{1}{2} \operatorname{ar}( ABC )$

$\Delta ABC$ is a triangle right-angled at $C$. $A$ line through the mid-point $M$ of hypotenuse $AB$ and parallel to $BC$ intersects $AC$ at $D$. Show that $D$ is the mid-point of $AC$.

$P, Q, R$ and $S$ are respectively the mid-points of sides $AB, BC, CD$ and $DA$ of quadrilateral $ABCD$ in which $AC = BD$ and $AC \perp BD$. Prove that $PQRS$ is a square.

In $\Delta ABC$,$D$,$E$,and $F$ are respectively the mid-points of sides $AB$,$BC$,and $CA$. Show that $\Delta ABC$ is divided into four congruent triangles by joining $D$,$E$,and $F$.