$\Delta ABC \sim \Delta DEF$ for the correspondence $ABC \leftrightarrow DEF$. If $3 AB = 5 DE$ and $DF = 9$,then $AC = \ldots$

In an isosceles right-angled triangle, the length of the hypotenuse is $24$. Find the area of the triangle.

Sides $AB$ and $AC$ and median $AD$ of a triangle $ABC$ are respectively proportional to sides $PQ$ and $PR$ and median $PM$ of another triangle $PQR$. Show that $\Delta ABC \sim \Delta PQR$.

In $\Delta PQR$,$m \angle Q = 90^{\circ}$ and $\overline{QM}$ is an altitude to the hypotenuse $PR$. If $QR = 9$ and $PR = 13.5$,find $RM$.

$S$ and $T$ are points on sides $PR$ and $QR$ of $\Delta PQR$ such that $\angle P = \angle RTS$. Show that $\Delta RPQ \sim \Delta RTS$.

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.