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

If $\Delta ABC \sim \Delta PQR$ for the correspondence $ABC \leftrightarrow PQR$,and given $AB + BC = 12$,$PQ + QR = 15$,and $AC = 8$,find the length of $PR$.

In $\Delta ABC$ and $\Delta XYZ$,$m\angle A = m\angle X$,$m\angle B = m\angle Y$ and $\frac{AB}{XY} = \frac{2}{3}$. If $AC = 7.2$,then $XZ = \ldots$

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $AC = 14\sqrt{2}$. If $AB = BC$,find the area 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.

In $\Delta ABC$,$\overline{AD}$ is a median. The bisectors of $\angle ADB$ and $\angle ADC$ intersect $\overline{AB}$ and $\overline{AC}$ at $E$ and $F$ respectively. Prove that $\overline{EF} \parallel \overline{BC}$.