In $\Delta PQR$,$\angle P$ is a right angle and $\overline{PX}$ is an altitude. If $QX = 4$ and $RX = 5$,then $PQ = \ldots \ldots$

$A$ $15 \, m$ long bamboo leans on a wall to reach the height of $12 \, m$ on the wall. Then,the lower end of the bamboo is $\ldots \ldots \, m$ away from the base of the wall.

In $\Delta ABC,$ $\overline{AD}$ is a median. If $AD=7,$ $AC=7$ and $BD=4,$ find the perimeter of $\Delta ABC.$

In $\square XYZW$,$\overline{XY} \parallel \overline{ZW}$ and $\overline{XZ} \cap \overline{YW} = \{P\}$. If $PX = 3x - 1$,$PY = 2x + 1$,$PZ = 5x - 3$,and $PW = 6x - 5$,find the value of $x$.

The lengths of the diagonals of a rhombus are $16 \, cm$ and $12 \, cm$. Then,the length of the side of the rhombus is (in $cm$)

In $\Delta ABC$,$D$ is the midpoint of $\overline{BC}$. $A$ line passing through $G$ (the centroid) intersects $\overline{AB}$ at $M$ and $\overline{AC}$ at $N$. If $3 AN = 2 AC$,prove that $AM = 2 MB$.