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

In $\Delta GBS$,$m\angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude to the hypotenuse $\overline{GS}$. Prove that $\frac{GB^{2}}{BS^{2}} = \frac{GM}{SM}$.

Given $\Delta ABC \sim \Delta PQR$ for the correspondence $ABC \leftrightarrow PQR.$ $\overline{AD}$ is a median in $\Delta ABC$ and $\overline{PM}$ is a median in $\Delta PQR.$ Prove that $\frac{AD}{PM} = \frac{AB}{PQ}.$

In the figure,$ABC$ is a triangle right-angled at $B$ and $BD \perp AC$. If $AD = 4 \, cm$ and $CD = 5 \, cm$,find $BD$ and $AB$.

Areas of two similar triangles are $36\, cm^{2}$ and $100\, cm^{2}$. If the length of a side of the larger triangle is $20\, cm$,find the length of the corresponding side of the smaller triangle. (in $cm$)

In $\Delta ABC$, $M$ and $N$ are the midpoints of $\overline{AB}$ and $\overline{AC}$ respectively. If the area of $\Delta ABC$ is $90$, find the area of $\Delta AMN$.