Question

If the sides $a,b,c$ of a triangle $ABC$ form successive terms of G.P. with common ratio $r(>1)$ then which of the following is correct?

• A. $A>\frac{\pi }{3}$
• B. $B\ge \frac{\pi }{3}$
• C. $C<\frac{\pi }{3}$
• D. $A<B<\frac{\pi }{3}$

Answer 1

The correct option is D $A<B<\frac{\pi }{3}$

let the sides of the triangle be $a,ar,a{r}^2$ and $r>1$
$\therefore a+ar>a{r}^2$
$\cos C=\frac{a^2+a^2{r}^2-a^2{r}^4}{2a^{2}r}=\frac{1+r^2-r^4}{2r}<\frac{1}{2}\Rightarrow C>\frac{\pi }{3}$
Also $\cos B=\frac{a^2-a^2{r}^2+a^2{r}^4}{2a^2{r}^2}$
$=\frac{1-r^2+r^4}{2r^2}>\frac{1}{2}\Rightarrow B<\frac{\pi }{3}$
Also, $a<ar<a{r}^2\Rightarrow A<B<CA<B<\frac{\pi }{3}<C$