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