Question

Mark the correct alternative in each of the following:
If the sides of a triangle are in the ratio \[1: \sqrt{3}: 2\] then the measure of its greatest angle is 

Options

• \[\frac{\pi}{6}\] 

• \[\frac{\pi}{3}\] 

• \[\frac{\pi}{2}\] 

• \[\frac{2\pi}{3}\]

Answer 1

Let ∆ABC be the given triangle such that its sides are in the ratio \[1: \sqrt{3}: 2\] 

\[\therefore a = k, b = \sqrt{3}k, c = 2k\]

Now,

\[a^2 + b^2 = k^2 + 3 k^2 = 4 k^2 = c^2\] 

So, ∆ABC is a right triangle right angled at C. 

\[\therefore C = 90°\]  

Using sine rule, we have 

\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]
\[ \Rightarrow \frac{k}{\sin A} = \frac{\sqrt{3}k}{\sin B} = \frac{2k}{\sin90°\] 
\[ \Rightarrow \sin A = \frac{1}{2} \text{ and } \sin B = \frac{\sqrt{3}}{2}\]
\[ \Rightarrow A = 30° a\text{ and } B = 60°\] 

Thus, the measure of its greatest angle is \[\frac{\pi}{2}\]  

Hence, the correct answer is option (c).