Question

If the sides of a triangle are in the ratio $1:\sqrt{3}:2$, then the angles of the triangle are in the ratio

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

Answer 1

The correct option is A 1 : 2 : 3

Let ABC be a $\Delta$ such that $a:b:c=1:\sqrt{3}:2$, we have
$\frac{sinA}{a}=\frac{sinB}{b}=\frac{sinC}{c}\Rightarrow \frac{sinA}{1}=\frac{sinB}{\sqrt{3}}=\frac{sinC}{2}\Rightarrow \frac{sinA}{\frac{1}{2}}=\frac{sinB}{\frac{\sqrt{3}}{2}}=\frac{sinC}{1}$
$=\frac{sinA}{sin\left(\frac{\pi }{6}\right)}=\frac{sinB}{sin\left(\frac{\pi }{3}\right)}=\frac{sinC}{sin\left(\frac{\pi }{2}\right)}$
$\therefore A:B:C=\frac{\pi }{6}:\frac{\pi }{3}:\frac{\pi }{2}=A:B:C=1:2:3$