If the Angles of a Triangle are in A.P., Then the Measure of One of the Angles in Radians is

Question:

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

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

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

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

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

Solution

Let the three angles of the triangle in A.P. be:

\[ (a-d), \ a, \ (a+d) \]

Since the sum of angles of a triangle is \(180^\circ\),

\[ (a-d) + a + (a+d) = 180^\circ \]

\[ 3a = 180^\circ \]

\[ a = 60^\circ \]

Thus, one angle is always:

\[ 60^\circ \]

Converting into radians:

\[ 60^\circ = \frac{60\pi}{180} \]

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

Hence, the correct option is:

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