Find the value of $x$ for which $DE \parallel AB$ in the given figure.

The perimeter of an equilateral triangle is $12$. Then,its area is............

$\Delta ABC \sim \Delta DEF$ for the correspondence $ABC \leftrightarrow FDE$. If $m \angle E + m \angle F = 130^{\circ}$,then $m \angle B = \dots$ (in $^{\circ}$)

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude to the hypotenuse $AC$. Then,$AB$ is the geometric mean of.......

In $\square ABCD$,$P \in \overline{AB}$,$Q \in \overline{BC}$,$R \in \overline{CD}$,and $S \in \overline{DA}$ such that $\frac{AP}{PB} = \frac{AS}{SD}$ and $\frac{CB}{QB} = \frac{CR}{RD}$. Prove that $\overline{PS} \parallel \overline{QR}$.

In $\square ABCD$,$T$ is a point on $\overline{BC}$. The line segment $\overrightarrow{AT}$ intersects $\overline{BD}$ at $M$ and the extension of $\overrightarrow{DC}$ at $O$. Prove that $AM^{2} = MT \times MO$.