In $\Delta XYZ$,$X-S-Y$,$X-T-Z$ and $\overline{ST} \parallel \overline{YZ}$. If $XS = 4$,$SY = 4.5$ and $XT = 8$,find $XZ$.

In an isosceles right-angled triangle, the length of the hypotenuse is $20$. Find the perimeter and the area of the triangle.

In $\Delta ABC$,$m\angle A + m\angle C = m\angle B$. If $AB = 7$ and $BC = 24$,then $AC = \ldots$

In the figure,$A, B$ and $C$ are points on $OP, OQ$ and $OR$ respectively such that $AB \parallel PQ$ and $AC \parallel PR$. Show that $BC \parallel QR$.

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 $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude to the hypotenuse $AC$. If $AB = 7$ and $BC = 24$,find the length of $BM$.