In $\square ABCD$,$\overline{AB} \parallel \overline{CD}$ and $\overline{AC} \cap \overline{BD} = \{M\}$. If $MA = 8$,$MB = 12$ and $MC = 6$,find $MD$.

$O$ is the point of intersection of the diagonals $AC$ and $BD$ of a trapezium $ABCD$ with $AB \parallel DC$. Through $O$,a line segment $PQ$ is drawn parallel to $AB$ meeting $AD$ in $P$ and $BC$ in $Q$. Prove that $PO = QO$.

The lengths of the sides of a triangle are $m^{2}+n^{2}$,$2mn$,and $m^{2}-n^{2}$,where $m > n > 0$. Prove that the triangle is a right-angled triangle.

$\Delta ABC \sim \Delta DEF$ 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 $AC$. If $AB = 7$ and $BC = 24$,find the length of $BM$.

In $\Delta ABC$,$m\angle B = 90^\circ$ and $\overline{BM}$ is an altitude to the hypotenuse $\overline{AC}$. If $AM = 4CM$,prove that $AB = 2BC$.