In $\Delta XYZ$,$m\angle Y = 90^{\circ}$ and $M$ is the midpoint of $\overline{XZ}$. If $XY = 3$ and $YZ = 4$,then $YM = \ldots$

Find the altitude of an equilateral triangle of side $8 \, cm$. (in $cm$)

If $\Delta PQR \sim \Delta XYZ$ under the correspondence $PQR \leftrightarrow YZX$,and $m \angle P = 80^{\circ}$ and $m \angle Q = 40^{\circ}$,find $m \angle Z$. (in $^{\circ}$)

In $\Delta ABC$,$\overline{AD}$ is a median. Then,by Apollonius' theorem,$\ldots \ldots \ldots \ldots$ holds good.

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.

If $\Delta ABC \sim \Delta PQR$ under the correspondence $ABC \leftrightarrow PQR$. If $AB = 12$,$\text{Area}(\Delta ABC) = 36$,and $\text{Area}(\Delta PQR) = 64$,then $PQ = \ldots$