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

A. `3:4:5`
B. `4:5:6`
C. `5:6:7`
D. `7:8:9`

Answer

B

Explanation

Let the sides of the triangle be a-d,a and a+d, with `agtdgt0`. Clearly, a-d is the smallest and a+d is the largest side. So, A is the smallest angle and C is the largest angle. It is given that C=2A. Thus, the angles of the triangle are A,2A and `pi-3A`.
Applying the law of sines, we obtain
`(a-d)/(sinA)=(a)/(sin(pi-3A))=(a+d)/(sin2A)`
`rArr" "(a-d)/(sinA)=(a)/(sin3A)=(a+d)/(sin2A)`
`rArr" "(a-d)/(sinA)=(a)/(3sinA-4sin^(3)A)=(a+d)/(2sinAcosA)`
`rArr" "(a-d)/(1)=(a)/(3-4sin^(2)A)=(a+d)/(2cosA)`
`rArr" "3-4sin^(2)A=(a)/(a-d)and2cosA=(a+d)/(a-d)`
`rArr" "4cos^(2)A-1=(a)/(a-d)and2cosA=(a+d)/(a-d)`
`rArr" "((a+d)/(a-d))^(2)-1=(a)/(a-d)rArra=5d`.
Thus, the sides of the triangle are
a-d,a,a+d i.e. 4d,5d,6d.
Hence, the ratio of the sides of the triangle is
`4d:5d:6d" i.e. "4:5:6`.