Question

If the angles of a triangle are in A.P., then the measures of one of the angles in radians is

Options

• \[\frac{\pi}{6}\]

   

• \[\frac{\pi}{3}\]

   

• \[\frac{\pi}{2}\]

   

• \[\frac{2\pi}{3}\]

   

Answer 1

\[\frac{\pi}{3}\]
Let the angles of the triangle be

\[\left( a - d \right)^\circ, \left( a \right)^\circ \text{ and }\left( a + d \right)^\circ\]
Thus, we have:
\[a - d + a + a + d = 180\]
\[ \Rightarrow 3a = 180\]
\[ \Rightarrow a = 60\]
Hence, the angles are

\[\left( a - d \right)^\circ, \left( a \right)^\circ\text{ and }\left( a + d \right)^\circ\]

\[\left( 60 - d \right)^\circ, 60^\circ\text{ and }\left( 60 + d \right)^\circ\]
60° is the only angle which is independent of d.
∴ One of the angles of the triangle (in radians) = \[\left( 60 \times \frac{\pi}{180} \right)\]

\[\frac{\pi}{3}\]