The correspondence $ABC \leftrightarrow XYZ$ between $\Delta ABC$ and $\Delta XYZ$ is a similarity. If $4AB = 5XY$ and $BC = 12$,find $YZ$. (in $.6$)

$A$ vertical pole of length $6\, m$ casts a shadow $4\, m$ long on the ground and at the same time a tower casts a shadow $28\, m$ long. Find the height of the tower. (in $m$)

$\Delta PQR \sim \Delta XYZ$ for the correspondence $PQR \leftrightarrow XZY$. If $PQ = 8$,$XZ = 12$ and the area of $\Delta PQR$ is $24$,find the area of $\Delta XYZ$.

In parallelogram $ABCD$,$P$ is a point on side $BC$. The line segment $DP$ intersects the extension of side $AB$ at point $L$. Prove that: $(1) \frac{DP}{PL} = \frac{DC}{BL}$ and $(2) \frac{DL}{DP} = \frac{AL}{DC}$.

In a $\triangle PQR$,$PR^{2} - PQ^{2} = QR^{2}$ and $M$ is a point on side $PR$ such that $QM \perp PR$. Prove that $QM^{2} = PM \times MR$.

In $\Delta ABC$ and $\Delta XYZ$,$\angle A \cong \angle Y$ and $\angle B \cong \angle Z$. If $\frac{AC}{YX} = \frac{5}{7}$ and $AB = 7$,find $YZ$.