In $\Delta PQR$ and $\Delta XYZ$,if $\frac{PQ}{XY} = \frac{QR}{YZ} = \frac{PR}{XZ} = \frac{2}{5}$,then the ratio of their areas $\text{Area}(\Delta PQR) : \text{Area}(\Delta XYZ) = \ldots$

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude to the hypotenuse $\overline{AC}$. If $AM = 12$ and $CM = 3$,find $BM$.

$CD$ and $GH$ are respectively the bisectors of $\angle ACB$ and $\angle EGF$ such that $D$ and $H$ lie on sides $AB$ and $FE$ of $\Delta ABC$ and $\Delta EFG$ respectively. If $\Delta ABC \sim \Delta FEG,$ show that:
$(i) \frac{CD}{GH} = \frac{AC}{FG}$
$(ii) \Delta DCB \sim \Delta HGE$
$(iii) \Delta DCA \sim \Delta HGF$

If $\Delta ABC \sim \Delta MNP$ for the correspondence $ABC \leftrightarrow MNP,$ the perimeter of $\Delta ABC$ is $18$ and the perimeter of $\Delta MNP$ is $27.$ If $AB = 8,$ then $MN = \dots$

$\Delta ABC \sim \Delta PQR$ 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 $AC = \ldots$

Is it true to say that if in two triangles,an angle of one triangle is equal to an angle of another triangle and two sides of one triangle are proportional to the two sides of the other triangle,then the triangles are similar? Give reasons for your answer.