Question

One of the angle of quadrilatral has measure $\frac{2\pi }{9}$ radians and other three angles are in the ratio $3:5:8$, Find the measures in radian.

Answer 1

Let $\angle A=\frac{2\pi }{9}=\frac{2\pi }{9}\times \frac{{180}^o}{\pi }={40}^o$

Let measures of other angles be $3x,5x$ and $8x$.

Let, $\angle B=3x,\angle C=5x$ and $\angle D=8x$

We know that sum of four angles of an quadrilateral is ${360}^o.$

$\therefore$ $\angle A+\angle B+\angle C+\angle D={360}^o.$

$\Rightarrow$ ${40}^o+3x+5x+8x={360}^o.$

$\Rightarrow$ $16x={320}^o.$

$\Rightarrow$ $x={20}^o.$

Now

$\angle B=3x=3\times {20}^o={60}^o$

$\angle C=5x=5\times {20}^o={100}^o$

$\angle D=8x=8\times {20}^o={160}^o.$

Measure in radian,

$\angle B={60}^o\times \frac{\pi }{{180}^o}=\frac{\pi }{3}$

$\angle C={100}^o\times \frac{\pi }{{180}^o}=\frac{5\pi }{9}$

$\angle D={160}^o\times \frac{\pi }{{180}^o}=\frac{8\pi }{9}$