In $\Delta PQR$,$m \angle Q = 90^\circ$. If $PQ : QR = 1 : 1$,then $PQ : PR = \ldots$

In the figure,$l \parallel m$ and line segments $AB, CD$ and $EF$ are concurrent at point $P$. Prove that $\frac{AE}{BF} = \frac{AC}{BD} = \frac{CE}{FD}$.

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 $\Delta XYZ$,$X-S-Y$,$X-T-Z$ and $\overline{ST} \parallel \overline{YZ}$. If $XS = 4$,$XT = 8$,$SY = x-4$ and $TZ = 3x-19$,find the value of $x$.

In $\Delta PQR$,$\overline{PM}$ is an altitude. If $PM^2 = QM \times RM$,prove that $\Delta PQR$ is a right-angled triangle.

The perimeter of an equilateral triangle is $18$. Then,the length of its altitude is $\ldots \ldots \ldots$