$A$ triangle whose sides measure ............ is a right-angled triangle.

The two triangles in the figure are congruent using a congruence theorem. It is given that $OQ = OR$. Which of these conditions,along with the given condition,is sufficient to prove that the two triangles are congruent to each other?

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude to the hypotenuse $\overline{AC}$. If $AM - MC = 7$ and $AB^{2} - BC^{2} = 175$,then $AC = \ldots$

Point $O$ lies in the interior of $\Delta ABC$. The bisectors of $\angle AOB, \angle BOC$ and $\angle COA$ intersect $\overline{AB}, \overline{BC}$ and $\overline{CA}$ at $D, E$ and $F$ respectively. Prove that $AD \times BE \times CF = DB \times EC \times FA$.

In $\Delta ABC$,$m \angle A = 90^{\circ}$ and $\overline{AM}$ is an altitude to the hypotenuse $\overline{BC}$. If $BM = 6$ and $CM = 2$,find the perimeter of $\Delta ABC$.

In $\Delta PQR$,$m \angle Q = 90^{\circ}$ and $\overline{QM}$ is an altitude. If $PQ = 40$ and $QR = 30$,find the area of $\Delta QMR$.