In $\Delta ABC$,the bisector of $\angle A$ intersects $\overline{BC}$ at $D$. If $AB = 5.2$,$AC = 10.4$,and $BD = 3.8$,find $BC$.

$\Delta PQR \sim \Delta XYZ$ 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 $\square XYZW$,$\overline{XY} \parallel \overline{ZW}$ and $\overline{XZ} \cap \overline{YW} = \{P\}$. If $PX = 3x - 1$,$PY = 2x + 1$,$PZ = 5x - 3$,and $PW = 6x - 5$,find the value of $x$.

In the given figure,$BD$ and $CE$ intersect each other at the point $P$. Is $\triangle PBC \sim \triangle PDE$ ? Why?

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude to the hypotenuse $\overline{AC}$. If $BM = 12$ and $CM = 18$,find $AB$.

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