Two objects at the points $P$ and $Q$subtend
an angle of 30° at a point A. Lengths $AR = 20 m$ and
$AS = 10 m$ are measured from $A$at right angles to$AP$ and
$AQ$respectively. If $PQ$ subtends equal angles of 30°, at $R$
and $S$, then length of $PQ$ is

Since$PQ$subtends the same angle of 30° at
each of the points $A, R$ and $S$, the points $A, P, Q, R$and $S$lie
on a circle. (Fig. 27.41)

Also $AR\perp AP$ and $AS\perp AQ$ 

$\Rightarrow \mathrm{\angle }RAS={90}^{\circ }-\mathrm{\angle }RAQ=\mathrm{\angle }PAQ={30}^{\circ }$

As it is given

                                        $AR=20\mathrm{m},AS=10\mathrm{m}$

Then $PQ$ and $RS$ are chords of the same circle making an
angle of 30° at a point A on the circumference of the circle
and hence are equal in length.

Now from  $∆ SAR$

                                $\begin{array}{l}R{S}^2={20}^2+{10}^2-2\times 20\times 10\cos {30}^{\circ }\\ P{Q}^2=500-200\sqrt{3}\end{array}$

                                $\Rightarrow PQ=\sqrt{500-200\sqrt{3}}$.