$\Delta ABC \sim \Delta XYZ$ for the correspondence $ABC \leftrightarrow XYZ$. If $AB = 12, XY = 8$ and the area of $\Delta ABC$ is $81$,find the area of $\Delta XYZ$.

$D$ is a point on side $QR$ of $\triangle PQR$ such that $PD \perp QR$. Will it be correct to say that $\triangle PQD \sim \triangle RPD$? Why?

If in triangles $ABC$ and $DEF$,$\frac{AB}{DE} = \frac{BC}{FD}$,then they will be similar,when

In $\Delta ABC$,the bisector of $\angle A$ intersects $\overline{BC}$ at $D$. If $AB = 12$,$BD = 9$ and $BC = 21$,find $AC$.

In trapezium $\square ABCD$,$\overline{AB} \parallel \overline{CD}$ and $\overline{AC} \cap \overline{BD} = \{M\}$. Prove that the correspondence $MAB \leftrightarrow MCD$ between $\Delta MAB$ and $\Delta MCD$ is a similarity.

$A$ $15 \, m$ long bamboo leans on a wall to reach the height of $12 \, m$ on the wall. Then,the lower end of the bamboo is $\ldots \ldots \, m$ away from the base of the wall.