$\Delta ABC \sim \Delta PQR$ for the correspondence $ABC \leftrightarrow PQR$. If $AB : PQ = 3 : 4$ and the perimeter of $\Delta PQR$ is $24$,find the perimeter of $\Delta ABC$.

In the figure,$D$ is a point on the hypotenuse $AC$ of $\Delta ABC,$ such that $BD \perp AC,$ $DM \perp BC,$ and $DN \perp AB.$ Prove that $DN^{2} = DM \cdot AN.$

Diagonals $AC$ and $BD$ of a trapezium $ABCD$ with $AB \parallel DC$ intersect each other at the point $O$. Using a similarity criterion for two triangles,show that $\frac{OA}{OC} = \frac{OB}{OD}$.

In $\Delta ABC$ and $\Delta XYZ$,$m \angle A = m \angle X$ and $m \angle B = m \angle Y$. $\overline{AM}$ is a median of $\Delta ABC$ and $\overline{XP}$ is a median of $\Delta XYZ$. Prove that $AM \times YZ = XP \times BC$.

$\square ABCD$ is a trapezium in which $\overline{ AD } \| \overline{ BC }, \overline{ AC } \cap \overline{ BD }=\{ P \} .$ If $PD =9, PA =5$ and $PB =7.2,$ then $AC =\ldots \ldots \ldots \ldots$

In the figure,if $\angle ACB = \angle CDA$,$AC = 8 \, cm$ and $AD = 3 \, cm$,find $BD$ (in $cm$).