Question

If the sides of a triangle are in G.P., and its larger angle is twice the smallest, then the common ratio r stisfies the inequality

• A.
• B.
• C.
• D. none of these

Answer 1

The correct option is B

Let the sides of the triangle be a/r, a and ar,with a>0 and r> 1. Let $\propto$ be the smallest angle, so that the largest angle is $2\propto$ . Then $\propto$ is opposite to the side a/r , and $2\propto$ is positive to the side ar. Applying sine rule, we get

$\frac{a/r}{sin\alpha }=\frac{ar}{sin2\alpha }$

$\Rightarrow \frac{sin2\alpha }{sin\alpha }$

$\Rightarrow 2cos\alpha =r^2$

$\Rightarrow r^2<2$

$\Rightarrow r<\sqrt{2}$