In $\Delta PQR$,$m\angle Q = 90^\circ$. If $PQ = 8$ and $QR = 15$,then $PR = \ldots$

In $\Delta ABC$,$m\angle B = 90^\circ$ and $\overline{BM}$ is an altitude to the hypotenuse $\overline{AC}$. Prove the following:
$(i)$ The correspondence $AMB \leftrightarrow ABC$ is a similarity.
(ii) The correspondence $BMC \leftrightarrow ABC$ is a similarity.
(iii) The correspondence $AMB \leftrightarrow BMC$ is a similarity.

In $\Delta PQR$,the bisector of $\angle P$ intersects $\overline{QR}$ at $S$. If $PQ = 8$,$QS = 5.6$ and $QR = 12.6$,find $PR$.

For the correspondence $ABC \leftrightarrow RPQ$ between $\Delta ABC$ and $\Delta PQR$,$\angle B$ corresponds to $\ldots \ldots \ldots \ldots$

In $\Delta ABC$,$\overline{AD}$ is a median. If $AB = 8$,$AC = 15$ and $AD = 8.5$,find $BC$.

The area of an equilateral triangle is $36 \sqrt{3}$. Find the measure of its sides.