In $\Delta ABC$,$A-M-B$,$A-N-C$ and $\overline{MN} \parallel \overline{BC}$. If $AM = 5$,$MB = 8$ and $AN = 10$,then $NC = \dots$

In $\Delta ABC,$ the bisector of $\angle A$ intersects $\overline{BC}$ at $D.$ If $AC = 4.2,$ $DC = 3,$ and $BC = 5,$ find $AB.$

Sides of triangles are given below. Determine which of them are right triangles. In case of a right triangle,write the length of its hypotenuse.
$13\, cm, 12\, cm, 5\, cm$

In $\Delta ABC$,$m \angle B = 90^\circ$ and $\overline{BM}$ is an altitude to the hypotenuse $AC$. If $AM = BM = 8$,then $AC = \ldots$

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

In $\Delta PQR$,$m \angle Q = 90^{\circ}$ and $\overline{QS}$ is an altitude. If $PS - SR = 10$ and $PQ^2 - QR^2 = 260$,then $PR = \dots$