If the sides `a , b , c` of a triangle `A B C` form successive terms of G.P. with common ratio `r(>1)` then which of the following is correct? (a) `A >pi/3 ' (b) `B≥π/3` '(c) `C<π/3` '(d) `A < B < π/3`

Options

A. `A gt pi//3`
B. `B ge pi//3`
C. `C lt pi//3`
D. `A lt B lt pi//3`

Answer

D

Explanation

Let the sides of the triangle be a, ar, `ar^(2)`
`:' ar^(2)` is the longest side `(r gt 1)`
`:. A + ar gt ar^(2)`
So, `cos C = (a^(2) + a^(2) r^(2) - a^(2) ^(4))/(2a^(2)r) = (1 + r^(2) -r^(4))/(2r) lt (1)/(2)`
`rArr C gt (pi)/(3)`
Also, `cosB = (a^(2) + a^(2) r^(4) -a^(2) r^(2))/(2a^(2) r^(2))`
`= (1 + r^(4) -r^(2))/(2r^(2))`
`= (1)/(2) [r^(2) + (1)/(r^(2)) -1]`
`= (1)/(2) [(r - (1)/(r))^(2) + 1] gt 1//2`
`rArr B lt pi//3`
Also, `a lt ar lt ar^(2)`
`rArr A lt B lt C`
`rArr A lt B lt (pi)/(3) lt C`