Corresponding sides of two similar triangles are in the ratio of $2:3$. If the area of the smaller triangle is $48 \, cm^2$,find the area of the larger triangle (in $cm^2$).

The areas of two similar triangles are $9$ and $16$. Then,the ratio of their corresponding sides is...............

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude to the hypotenuse $\overline{AC}$. Then,$\angle BAM \cong \ldots$

In $\Delta ABC$,$m\angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude to the hypotenuse $\overline{AC}$. If $AC = 25$ and $AM = 16$,then $BM = \dots$

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude to the hypotenuse $\overline{AC}$. If $AB = 2\sqrt{10}$ and $AM = 5$,find $CM$.

In the figure,$\Delta ABC$ and $\Delta AMP$ are two right-angled triangles,right-angled at $B$ and $M$ respectively. Prove that:
$(i)$ $\Delta ABC \sim \Delta AMP$
$(ii)$ $\frac{CA}{PA} = \frac{BC}{MP}$