If $\Delta ABC \sim \Delta PQR$ for the correspondence $ABC \leftrightarrow PQR$,and given $AB + BC = 12$,$PQ + QR = 15$,and $AC = 8$,find the length of $PR$.

State which pairs of triangles in the figure are similar. Write the similarity criterion used by you for answering the question and also write the pairs of similar triangles in the symbolic form.

In the figure,$ABC$ and $DBC$ are two triangles on the same base $BC$. If $AD$ intersects $BC$ at $O$,show that $\frac{\operatorname{ar}(ABC)}{\operatorname{ar}(DBC)} = \frac{AO}{DO}$.

In triangles $ABC$ and $DEF$,$\angle B = \angle E$,$\angle F = \angle C$ and $AB = 3 DE$. Then,the two triangles are

State which pairs of triangles in the figure are similar. Write the similarity criterion used by you for answering the question and also write the pairs of similar triangles in the symbolic form.

In $\Delta ABC$,$m \angle B = 90^\circ$ and $\overline{BM}$ is an altitude to the hypotenuse $AC$. If $BC = 2x + 2$,$CM = x + 1$,and $AC = 5x$,find the value of $x$.