Question

If the angles of a triangle are in the ratio $1:2:3$ the corresponding sides are in the ratio

• A. $2:3:1$
• B. $\sqrt{3}:2:1$
• C. $2:\sqrt{3}:1$
• D. $1:\sqrt{3}:2$

Answer 1

The correct option is D $1:\sqrt{3}:2$

Let $A=\theta ,B=2\theta ,C=3\theta$
$\because A+B+C=\pi$
$\Rightarrow \theta +2\theta +3\theta =\pi$
$\Rightarrow 6\theta =\pi$
$\Rightarrow \theta =\frac{\pi }{6}$
$\therefore A=\theta =\frac{\pi }{6}$
$B=2\theta =2\times \frac{\pi }{6}=\frac{\pi }{3}$
$C=3\theta =3\times \frac{\pi }{6}=\frac{\pi }{2}$
$\therefore a:b:c=\sin \frac{\pi }{6}:\sin \frac{\pi }{3}:\sin \frac{\pi }{2}$
$\Rightarrow a:b:c=\frac{1}{2}:\frac{\sqrt{3}}{2}:1$
or $a:b:c=1:\sqrt{3}:2$