$\Delta ABC \sim \Delta PQR$ for the correspondence $ABC \leftrightarrow PQR$. The perimeter of $\Delta ABC$ is $35$ and the perimeter of $\Delta PQR$ is $28$. If $PR = 4\sqrt{10}$,then $AC = \ldots$

$\Delta ABC \sim \Delta XYZ$ for the correspondence $ABC \leftrightarrow XYZ$. $\overline{AD} \perp \overline{BC}$,$D \in \overline{BC}$ and $\overline{XM} \perp \overline{YZ}$,$M \in \overline{YZ}$. Prove that $\frac{AD}{XM} = \frac{BC}{YZ}$.

For $\Delta ABC$ and $\Delta XYZ$,$\frac{AB}{XZ} = \frac{BC}{XY} = \frac{AC}{YZ}$. Then,the correspondence $ABC \leftrightarrow \ldots \ldots$ between them is a similarity.

The ratio of the corresponding sides of two similar triangles is $4:9$. Then,the ratio of their areas is $\ldots \ldots$.

From an airport,two planes begin their journey at the same instant to fly at the same height. One plane moves towards north at an average speed of $800 \, km/h$ and the other plane moves towards east at an average speed of $840 \, km/h$. Find the distance in $km$ between those planes after $1 \frac{1}{2}$ hours of journey.

In $\Delta ABC$,$m \angle A = 90^{\circ}$ and $m \angle B = 30^{\circ}$. Then,which of the following holds true?