Question

The angles of a quadrilateral are in A.P. and the greatest angle is

${120}^{\circ }$. Express the angles in radians.

Answer 1

Let the angles in degrees be a - 3d, a - d, a + d, a + 3d

Then,

Sum of the angles $={360}^{\circ }$

$\Rightarrow 4a={360}^{\circ }a={90}^{\circ }$

Also,

greatest angle $={120}^{\circ }$

a + 3d = ${120}^{\circ }$

$\Rightarrow {90}^{\circ }+3d={120}^{\circ }\Rightarrow 3d={30}^{\circ }$

$\Rightarrow d={10}^{\circ }$

Hence, angles in degrees

${60}^{\circ },{80}^{\circ },{100}^{\circ },{120}^{\circ }$

and in radians, we know that

$1^{\circ }={\left(\frac{\pi }{180}\right)}^c\therefore 60\times \frac{\pi }{180}=\frac{\pi }{3},80\times \frac{\pi }{180}=\frac{4\pi }{9},100\times \frac{\pi }{180}=\frac{5\pi }{9}and120\times \frac{\pi }{180}=\frac{2\pi }{3}\therefore \frac{\pi }{3},\frac{4\pi }{9},\frac{5\pi }{9},\frac{2\pi }{3}$