Question

If the lengths of the sides of a triangle are in A.P. and the greatest angle is double the smallest, then a ratio of lengths of the sides of this triangle is ______.

Options

• 5 : 9 : 13

• 4 : 5 : 6

• 3 : 4 : 5

• 5 : 6 : 7

Answer 1

If the lengths of the sides of a triangle are in A.P. and the greatest angle is double the smallest, then a ratio of lengths of the sides of this triangle is 4 : 5 : 6.

Explanation:

Let the sides of triangle be a > b > c where

Given A = 2C

∵ A + B + C = π and A = 2C

`\implies` B = π – 3C   ...(i)

∵ a, b, c are in A.P. `\implies` a + c = 2b

`\implies` sin A + sin C = 2 sin B  ...(ii)

`\implies` sin A = sin (2C) and sin B = sin 3C

From (ii),

sin 2C + sin C = 2 sin 3C

`\implies` (2cos C + 1) sin C = 2 sin C (3 – 4 sin²C)

`\implies` 2cos C + 1 = 6 – 8 (1 – cos²C)

`\implies` 8cos²C – 2cos C – 3 = 0

`\implies` cos C = `3/4` or cos C = `-1/2`

∵ C is acute angle

`\implies` cos C = `3/4 \implies` sin C = `sqrt(7)/4`

and sin A = 2 sin C cos C = `2 xx sqrt(7)/4 xx 3/4 = (3sqrt(7))/8`

sin B = `(3sqrt(7))/4 - (4sqrt(7))/4 xx 7/16 = (5sqrt(7))/16`

`\implies` sin A : sin B : sin C : : a : b : c is 6 : 5 : 4