Question

The angles of a quadrilateral are in A.P. and the greatest angle is 120^o, the angles in radians are

• A. $\frac{\pi}{3},\frac{4\pi}{9},\frac{5\pi}{9},\frac{2\pi}{3}$
• B. $\frac{\pi}{3},\frac{\pi}{2},\frac{2\pi}{3},\frac{3\pi}{3}$
• C. $\frac{5\pi}{18},\frac{8\pi}{18},\frac{11\pi}{18},\frac{12\pi}{18}$
• D. None of these

Answer 1

Answer: (A)

$\frac{\pi}{3},\frac{4\pi}{9},\frac{5\pi}{9},\frac{2\pi}{3}$

Answer 2

Let the angles in degrees be $\alpha - 3\delta,\alpha - \delta,\alpha + \delta,\alpha + 3\delta$

Sum of the angles $= 4\alpha = 360^{o}$ ∴ $\alpha = 90^{o}$

Also greatest angle $= \alpha + 3\delta = 120^{o}$,

Hence, $3\delta = 120^{o} - \alpha = 120^{o} - 90^{o} = 30^{o}$ ∴ $\delta = 10^{o}$

Hence the angles are $90^{o} - 30^{o},90^{o} - 10^{o},90^{o} + 10^{o}$ and

$90^{o} + 30^{o}$

That is, the angles in degrees are $60^{o},80^{o},100^{o}$ and $120^{o}$

∴ In terms of radians the angles are

$60 \times \frac{\pi}{180},80 \times \frac{\pi}{180},100 \times \frac{\pi}{180}$and $120 \times \frac{\pi}{180}$ that is

$\frac{\pi}{3},\frac{4\pi}{9},\frac{5\pi}{9}$ and $\frac{2\pi}{3}$.