In $\Delta ABC$,$m \angle B = 90^{\circ}$,$AC - BC = 2$ and $BC - AB = 7$. Find the perimeter of $\Delta ABC$.

In $\Delta ABC$,$D \in \overline{AB}$,$E \in \overline{AC}$ and $\overline{DE} \parallel \overline{BC}$. $F$ is a point on $\overline{AD}$ such that $\overline{EF} \parallel \overline{CD}$. Prove that $AD^2 = AB \times AF$.

$D$ and $E$ are points on the sides $AB$ and $AC$ of a triangle $ABC$ respectively,such that $AD = 2 \, cm$,$BD = 3 \, cm$,$BC = 7.5 \, cm$,and $DE \parallel BC$. Find the length of $DE$ (in $cm$).

In $Fig.$,line segment $DF$ intersects the side $AC$ of a triangle $ABC$ at the point $E$ such that $E$ is the mid-point of $CA$ and $\angle AEF = \angle AFE$. Prove that
$\frac{BD}{CD} = \frac{BF}{CE}$

In $\Delta PQR$,$\overline{QM}$ is an altitude. If $PQ = 2PM$ and $RM = 3PM$,prove that $\Delta PQR$ is a right-angled triangle.

In $\square ABCD$,$AB = AD$. The bisector of $\angle BAC$ intersects $\overline{BC}$ at $E$ and the bisector of $\angle DAC$ intersects $\overline{DC}$ at $F$. Prove that $\overline{EF} \parallel \overline{BD}$.