Question

Let $PQRS$ be a quadrilateral in a plane, where $QR =1, \angle PQR =\angle QRS =70^{\circ}, \angle PQS =15^{\circ}$ and $\angle PRS =40^{\circ}$. If $\angle RPS =\theta^{\circ}, PQ =\alpha$ and $PS =\beta$, then the interval(s) that contain(s) the value of $4 \alpha \beta \sin \theta^{\circ}$ is/are

• A. $(0, \sqrt{2})$
• B. $(1,2)$
• C. $(\sqrt{2}, 3)$
• D. $(2 \sqrt{2}, 3 \sqrt{2})$

Answer 1

Answer: (A), (B)

$ \angle PRQ =70^{\circ}-40^{\circ}=30^{\circ}$
$ \angle RQS =70^{\circ}-15^{\circ}=55^{\circ} $
$ \angle QSR =180^{\circ}-55^{\circ}-70^{\circ}=55 $
$ \therefore QR = RS =1$
$ \angle QPR =180^{\circ}-70^{\circ}-30^{\circ}=80^{\circ}$

Apply sine-rule in $\triangle PRQ$ :
$\frac{\alpha}{\sin 30^{\circ}}=\frac{1}{\sin 80^{\circ}} \Rightarrow \alpha=\frac{1}{2 \sin 80^{\circ}} .....$(1)
Apply sine-rule in $\triangle PRS$
$ \frac{\beta}{\sin 40^{\circ}}=\frac{1}{\sin \theta} \Rightarrow \beta \sin \theta=\sin 40^{\circ}.....$(2)
$4 \alpha \beta \sin \theta=\frac{4 \sin 40^{\circ}}{2 \sin 80^{\circ}}=\frac{4 \sin 40^{\circ}}{2\left(2 \sin 40^{\circ} \cos 40^{\circ}\right)}$
$=\sec 40^{\circ} $
Now $\sec 30^{\circ} < \sec 40^{\circ} < \sec 45^{\circ} $
$ \Rightarrow \frac{2}{\sqrt{3}} < \sec 40^{\circ} < \sqrt{2}$