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 :

• A. 5 : 9 : 13
• B. 5 : 6 : 7
• C. 4 : 5 : 6
• D. 3 : 4 : 5

Answer 1

Answer: (C)

$a<b<c$ are in A.P.
$\angle C=2\angle A$ (Given)
$\Rightarrow$ sinC = sin 2 A
$\Rightarrow$ sin C = 2 sin A. cos A
$\Rightarrow \frac{\sin C}{\sin A}=2\cos A$
$\Rightarrow \frac{c}{a}=2\frac{b^2+c^2-a^2}{2bc}$
Put $a=b-\lambda ,c=b+\lambda ,\lambda >0$
$\Rightarrow \lambda =\frac{b}{5}$
$\Rightarrow a=b-\frac{b}{5}=\frac{4}{5}b,c=b+\frac{b}{5}=\frac{6b}{5}$
$\Rightarrow$ required ratio = $4:5:6$