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

Prove that the sum of the squares of the sides of a rectangle is equal to the sum of the squares of its diagonals.

The length of the hypotenuse of an isosceles right-angled triangle is $8 \, cm$. Find the length of the other two sides.

$A$ 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 $4\, seconds$.

When a ladder leans on a wall to reach a height of $8 \, m$, its lower end rests $6 \, m$ away from the base of the wall. If the same ladder leans on the wall to reach a height of $6 \, m$, find the distance of its lower end from the base of the wall. (in $m$)

Point $O$ lies in the interior of $\Delta PQR$. Given $O-A-P$,$O-B-Q$,$O-C-R$,$\overline{AB} \parallel \overline{PQ}$ and $\overline{BC} \parallel \overline{QR}$. Prove that $\overline{AC} \parallel \overline{PR}$.