If the sides of a triangle are in GP and its largest angle is twice tha smallset then the common ratio r satisfies the inequality

Options

A. `0ltrltsqrt(2)`
B. `1ltrltsqrt(2)`
C. `1ltrlt2`
D. none of these

Answer

B

Explanation

Let the sides of the triangle be `a//r`, a and ar, with `agt0gtandrgt1`.
Let `alpha` be the smallest angle, so that the largest angle is `2alpha`. Then, `alpha` is opposite to the side `a//r and2alpha` is opposite to the side ar. Applying sine rule, we get
`(a//r)/(sinalpha)=(ar)/(sin2alpha)`
`rArr" "(sin2alpha)/(sinalpha)=r^(2)`
`rArr2cosalpha=r^(2)`
`rArr" "r^(2)lt2" "[becausealpha!=:.2cosalphalt2]`
`rArr" "rltsqrt(2)`
Hence, `1ltrltsqrt(2)`