Question

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

• A. $1:3:5$
• B. $2:3:4$
• C. $3:2:1$
• D. $1:2:3$

Answer 1

Answer: (D)

$1:2:3$

Solution:

Let

$a=x,b=\sqrt{3}x,c=2x$

we have

Cosine rule $\Rightarrow \cos C=\frac{a^2+b^2-c^2}{2ab}$

$cosC=\frac{x^2+(\sqrt{3}x{)}^2-(2x{)}^2}{2x\left(\sqrt{3}x\right)}$

$C={\cos }^{-1}\left(0\right)=\frac{\pi }{2}={90}^{\circ }$

similarly $cosB=\frac{x^2+(2x{)}^2-(\sqrt{3}x{)}^2}{2\left(2x\right)\left(x\right)}$

$B={\cos }^{-1}\left(\frac{1}{2}\right)={60}^{\circ }$

we have $A+B+C=180$

$A=180-B-C$

$A=180-60-90=30$

$A:B:C=30:60:90\Rightarrow A:B:C=1:2:3$