In $\Delta ABC$,$A-M-B$,$A-N-C$ and $\overline{MN} \parallel \overline{BC}$. If $AM = 3x - 1$,$MB = 2x + 1$,$AN = 3x + 1$ and $NC = 5x + 1$,find the value of $x$.

$A$ girl of height $90\, cm$ is walking away from the base of a lamp-post at a speed of $1.2\, m/s$. If the lamp is $3.6\, m$ above the ground,find the length of her shadow after $4\, seconds$.

$ABCD$ is a trapezium with $AB \parallel DC.$ $E$ and $F$ are points on non-parallel sides $AD$ and $BC$ respectively such that $EF \parallel AB$ (see Figure). Show that $\frac{AE}{ED} = \frac{BF}{FC}$.

In $\Delta ABC$,if $\ldots \ldots \ldots \ldots$,then by Apollonius' theorem,$AB^{2} + AC^{2} = 2(AD^{2} + BD^{2})$ holds good.

In $\Delta ABC$,$A-M-B$,$A-N-C$ and $\overline{MN} \parallel \overline{BC}$. If $AM : AB = 5 : 7$ and $AN = 4$,find $NC$.

$O$ is any point inside a rectangle $ABCD$ (see Figure). Prove that $OB^{2} + OD^{2} = OA^{2} + OC^{2}$.