The median of a triangle divides it into two:

$(1)$ Area of a rhombus $= \frac{1}{2} \times \ldots \ldots \ldots$
$(2)$ Area of a triangle $= \ldots \ldots \ldots$

Write True or False and justify your answer:
$ABC$ and $BDE$ are two equilateral triangles such that $D$ is the mid-point of $BC.$ Then $\operatorname{ar}(\triangle BDE) = \frac{1}{4} \operatorname{ar}(\triangle ABC).$

In $\Delta ABC$,$\angle B = 90^{\circ}$,$AB = 8\, \text{cm}$ and $BC = 15\, \text{cm}$,then $\text{ar}(\Delta ABC) = \dots \text{cm}^2$.

$ABCD$ is a parallelogram. If $\operatorname{ar}(ABC) = 42 \, \text{cm}^2$,then find $\operatorname{ar}(ABCD)$ in $\text{cm}^2$.

$PQRS$ is a square. $T$ and $U$ are the mid-points of $PS$ and $QR$ respectively. Find the area of $\Delta OTS$,if $PQ = 8 \, cm$,where $O$ is the point of intersection of $TU$ and $QS$.