Question

A variable triangle is inscribed in a circle of radius R. If the rate of change of side is R times the rate of change of the opposite angle, then that angle is

• A. $\pi /6$
• B. $\pi /4$
• C. $\pi /3$
• D. $\pi /2$

Answer 1

Answer: (C)

$\pi /3$

By sine rule
$\frac{a}{sinA}=\frac{b}{sinB}=\frac{c}{sinc}=2R$.
Hence
$a=2Rsin\theta$. ...(i)
Now it is given that
$\frac{d\left(a\right)}{dt}=R.\frac{d\left(\theta \right)}{dt}$. (ii)
Differentiating i with respect to time we get
$\frac{da}{dt}=2R.\frac{d\left(sin\theta \right)}{dt}$
Or
$R.\frac{d\theta }{dt}=2R.cos\theta .\frac{d\theta }{dt}$
Or
$cos\theta =\frac{1}{2}$
Or
$\theta ={60}^0$
Hence $a=b=c={60}^0$
Therefore the triangle is an equilateral triangle.