Question

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

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

Answer 1

The correct option is D

$1:2:3$

Explanation for the correct option:

Step 1: Given data

$a:b:c=1:\sqrt{3}:2$

Let the length of the sides $a=\lambda ,b=\sqrt{3}\lambda ,c=2\lambda$

Step 2: Apply cosine rule

Now, by cosine rule

$$\begin{gathered} \begin{array}{cc}& c^2=a^2+b^2-2bc\cos \left(C\right)\\ \Rightarrow & \cos \left(C\right)=\frac{\left(a^2+b^2-c^2\right)}{2ab}\\ \Rightarrow & \cos \left(C\right)=\frac{\left({\lambda }^2+3{\lambda }^2-4{\lambda }^2\right)}{2\sqrt{3}{\lambda }^2}\\ \Rightarrow & \cos \left(C\right)=0\end{array} \\ \Rightarrow C=90° \end{gathered}$$

Similarly, $\cos \left(B\right)=\frac{(a^2+c^2-b^2)}{2ca}$

$$\begin{gathered} \Rightarrow \cos \left(B\right)=\frac{{\lambda }^2+4{\lambda }^2-3{\lambda }^2}{4{\lambda }^2}=\frac{1}{2} \\ \Rightarrow B=60° \end{gathered}$$

Now, in the given triangle

$\begin{array}{cc}& \angle A+\angle B+\angle C=180°\\ \Rightarrow & \angle A+60+90=180\\ \Rightarrow & \angle A=30°\end{array}$

So, the ratio of angles is $30°:60°:90°=1:2:3$

Hence, option D is correct.