Question

The angles of a triangle are in A.P. such that the greatest is 5 times the least. Find the angles in radians.

Answer 1

Let the angles of the triangle be

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

\[a - d + a + a + d = 180\]
\[ \Rightarrow 3a = 180\]
\[ \Rightarrow a = 60\]
Given:

Greatest angle= 5 x Least angle
\[\text{ or,} \frac{\text{ Greatest angle }}{\text{ Least angle }} = 5\]
\[\text{ or, }\frac{a + d}{a - d} = 5\]
\[\text{ or, }\frac{60 + d}{60 - d} = 5\]
\[\text{ or, }60 + d = 300 - 5d\]
\[\text{ or, }6d = 240\]
\[\text{ or, }d = 40\]
Hence, the angles are

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

\[20^\circ, 60^\circ \text{ and }100^\circ\], respectively.
∴ Angles of the triangle in radians = \[\left( 20 \times \frac{\pi}{180} \right), \left( 60 \times \frac{\pi}{180} \right) \text{ and }\left( 100 \times \frac{\pi}{180} \right)\]

\[\frac{\pi}{9}, \frac{\pi}{3} \text{ and }\frac{5\pi}{9}\]