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

$\square ABCD$ is a trapezium in which $\overline{ AD } \| \overline{ BC }, \overline{ AC } \cap \overline{ BD }=\{ P \} .$ If $PD =9, PA =5$ and $PB =7.2,$ then $AC =\ldots \ldots \ldots \ldots$

It is given that $\triangle ABC \sim \triangle EDF$ such that $AB = 5 \, cm$,$AC = 7 \, cm$,$DF = 15 \, cm$,and $DE = 12 \, cm$. Find the lengths of the remaining sides of the triangles.

In $\Delta ABC$,$A-N-B$,$A-M-C$ and $B-X-C$. $\overline{XM} \parallel \overline{AB}$ and $\overline{XN} \parallel \overline{AC}$. $\overline{MN}$ intersects $\overline{CB}$ at $T$. Prove that $TX^2 = TB \times TC$.

In $\Delta ABC$,$AB + BC = 23$,$BC + AC = 32$,and $AB + AC = 25$. Show that $\Delta ABC$ is a right-angled triangle.

In $\Delta ABC$,$AB = AC$ and $m \angle A = 90^\circ$. If $BC = \sqrt{2} a$,then the area of $\Delta ABC$ is $\ldots \ldots \ldots \ldots$ $(a \in R^+)$.