Question

The angles of a triangle are in AP and the least angle is ${30}^o$. The greatest angle in radians is:

• A. $\frac{7\pi }{12}$
• B. $\frac{2\pi }{3}$
• C. $\frac{5\pi }{6}$
• D. $\frac{\pi }{2}$

Answer 1

Answer: (D)

$\frac{\pi }{2}$

Let triangle be $ABC$

and we have

$A={30}^o=\frac{\pi }{6}$

$\therefore B=\frac{\pi }{6}+d,C=\frac{\pi }{6}+2d$

Where, $d$ is the common difference of the A.P.

Now, sum of all the angles is $\pi$

$\Rightarrow \frac{\pi }{6}+\frac{\pi }{6}+\frac{\pi }{6}+3d=\pi$

$\Rightarrow \frac{\pi }{2}+3d=\pi \Rightarrow d=\frac{\pi }{6}$

Hence, greatest angle $=\frac{\pi }{6}+2d=\frac{\pi }{6}+2\times \frac{\pi }{6}=\frac{\pi }{2}$