Question

The angle of a quadrilateral are in A.P. and the greatest angle is 120°. Express the angles in radians.

Answer 1

Let the angles of the quadrilateral be $\left(a-3d\right)°,\left(a-d\right)°,\left(a+d\right)°\mathrm{and}\left(a+3d\right)°$.
We know:
$$\begin{gathered} a-3d+a-d+a+d+a-2d=360 \\ \Rightarrow 4a=360 \\ \Rightarrow a=90 \end{gathered}$$
We have:
Greatest angle = 120°
Now,
$$\begin{gathered} a+3d=120 \\ \Rightarrow 90+3d=120 \\ \Rightarrow 3d=30 \\ \Rightarrow d=10 \end{gathered}$$

Hence, $\left(a-3d\right)°,\left(a-d\right)°,\left(a+d\right)°\mathrm{and}\left(a+3d\right)°$ are $60°,80°,100°\mathrm{and}120°$, respectively.

Angles of the quadrilateral in radians = $\left(60\times {\displaystyle \frac{\mathrm{\pi }}{180}}\right),\left(80\times {\displaystyle \frac{\mathrm{\pi }}{180}}\right),\left(100\times {\displaystyle \frac{\mathrm{\pi }}{180}}\right)\mathrm{and}\left(120\times {\displaystyle \frac{\mathrm{\pi }}{180}}\right)$
=$\frac{\mathrm{\pi }}{3},\frac{4\mathrm{\pi }}{9},\frac{5\mathrm{\pi }}{9}\mathrm{and}\frac{2\mathrm{\pi }}{3}$