In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude to the hypotenuse $\overline{AC}$. If $AM = 9$ and $CM = 16$,find the perimeter of $\Delta ABC$.

In $\square ABCD$,$P \in \overline{AB}$,$Q \in \overline{BC}$,$R \in \overline{CD}$,and $S \in \overline{AD}$ such that $\frac{AP}{AB} = \frac{CQ}{BC} = \frac{CR}{CD} = \frac{AS}{AD} = \frac{1}{3}$. Prove that $\square PQRS$ is a parallelogram.

Prove that if in a triangle the square of one side is equal to the sum of the squares of the other two sides,then the angle opposite the first side is a right angle.

In $\Delta ABC$,the bisector of $\angle A$ intersects $\overline{BC}$ at $D$. If $AB = 12$,$AC = 8$ and $BD = 9$,find $DC$.

In rhombus $ABCD$,$AC = 16$ and $BD = 30$. Find its perimeter.

In $\square ABCD$,$\overline{AB} \parallel \overline{CD}$,$M \in \overline{AD}$ and $N \in \overline{BC}$. If $\overline{MN} \parallel \overline{AB}$,prove that $\frac{DM}{MA} = \frac{CN}{NB}$.