In $\Delta ABC$,$\angle B = 90^{\circ}$ and $BM$ is an altitude to the hypotenuse $AC$. If $AB = 12 \, cm$ and $BC = 16 \, cm$,then find the length of $BM$ in $cm$.

If the mid-points of the sides of a quadrilateral are joined in order,prove that the area of the parallelogram so formed will be half of the area of the given quadrilateral.

$ABCD$ is a trapezium with parallel sides $AB = a \text{ cm}$ and $DC = b \text{ cm}$. $E$ and $F$ are the mid-points of the non-parallel sides. The ratio of $\operatorname{ar}(ABFE)$ and $\operatorname{ar}(EFCD)$ is

In $\Delta PQR$,$\angle Q = 90^{\circ}$,$QR = 21 \text{ cm}$ and $PR = 29 \text{ cm}$,then find the area of $\Delta PQR$ in $\text{cm}^2$.

If the medians of a $\Delta ABC$ intersect at $G$,show that $\operatorname{ar}(AGB) = \operatorname{ar}(AGC) = \operatorname{ar}(BGC) = \frac{1}{3} \operatorname{ar}(ABC)$.

In the given figure,$ABCD$ is a trapezium with $AB \parallel DC$. $E$ is a point on extended $BC$. Prove that $ar(BDE) = ar(ACED)$.