Question

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

(a) $\frac{\mathrm{\pi }}{6}$

(b) $\frac{\mathrm{\pi }}{3}$

(c) $\frac{\mathrm{\pi }}{2}$

(d) $\frac{2\mathrm{\pi }}{3}$

Answer 1

(b) $\frac{\mathrm{\pi }}{3}$

Let the angles of the triangle be $\left(a-d\right)°,\left(a\right)°\mathrm{and}\left(a+d\right)°$.
Thus, we have:
$$\begin{gathered} a-d+a+a+d=180 \\ \Rightarrow 3a=180 \\ \Rightarrow a=60 \end{gathered}$$
Hence, the angles are $\left(a-d\right)°,\left(a\right)°\mathrm{and}\left(a+d\right)°$, i.e., $\left(60-d\right)°,60°\mathrm{and}\left(60+d\right)°$.
60° is the only angle which is independent of d.
∴ One of the angles of the triangle (in radians) =$\left(60\times {\displaystyle \frac{\mathrm{\pi }}{180}}\right)$= $\frac{\mathrm{\pi }}{3}$