If the lengths of the sides of a triangle are in AP and the greatest angle is double the smallest, then a ratio of lengths of the sides of this triangle isA. `3 : 4 : 5`B. `4 : 5 : 6`C. `5 : 9 : 13`D. `5 : 6 : 7`

If the lengths of the sides of a triangle are in AP and the greatest a

1.	If the lengths of the sides of a triangle are in AP and the greatest angle is double the smallest, then a ratio of lengths of the sides of this triangle isA. `3 : 4 : 5`B. `4 : 5 : 6`C. `5 : 9 : 13`D. `5 : 6 : 7`
Answer» Correct Answer - B Let a, b and c be the lengths of sides of a `DeltaABC` such that `a lt b lt c`. Since, sides are in AP. `therefore 2b = a + c " ...(i) "` Let `angleA = theta` Then, `angleC = 2 theta " [according to the equation]"` So, `angleB = pi - 3 theta " (ii)"` On applying sine rule in Eq. (i), we get 2 sin B = sin A + sin C `implies 2 sin (pi - 3 theta) = sin theta + sin 2 theta " [from Eq. (ii)]"` `implies 2 sin 3 theta = sin theta + sin 2 theta` `implies 2 [3sin theta - 4 sin^(3) theta]= sin theta + 2sin theta cos theta` `implies6 - 8sin^(2) theta = 1 + 2 cos theta" "[because sin theta " can not be zero]"` `implies 6 - 8(1 - cos^(2)theta) = 1 + 2cos theta` `implies 8 cos^(2) theta - 2 cos theta - 3 = 0` `implies (2 cos theta + 1)(4 cos theta - 3) = 0` `implies cos theta = (3)/(4)` or `cos theta = - (1)/(2)` (rejected). Clearly, the ratio of sides is a : b : c `=sin theta : sin 3 theta : sin 2 theta` `=sin theta : (3sin theta - 4 sin^(3) theta) : 2 sin theta cos theta` `=1 : (4 cos^(2) theta - 1) : 2 cos theta` `=1 : (5)/(4):(6)/(4) = 4 : 5: 6`