In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BD}$ is an altitude to the hypotenuse $AC$. If $AD = 9$ and $CD = 27$,find $AB$.

In $\Delta ABC$,the bisector of $\angle A$ intersects $\overline{BC}$ at $D$. Prove that $BD = \frac{BC \times AB}{AB + AC}$ and $DC = \frac{BC \times AC}{AB + AC}$.

In $\Delta ABC$,$A-M-B$,$A-N-C$ and $\overline{MN} \parallel \overline{BC}$. If $MN = 3$ and $BC = 7$,then $M$ divides $\overline{AB}$ from $A$ in the ratio.......

In $\Delta ABC$,$\overline{AB} \cong \overline{AC}$ and $\overline{AD}$ is a median. If $BC = 12$ and $AD = 8$,find $AB$.

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

In $\Delta PQR$,$PQ = 25$ and $QR = 24$. If $PR = \dots$,then $\angle R$ is a right angle.