Question

If the sides of an $∆ABC$ are in A. P. and the greatest angle is double the smallest. The ratio of the sides of $∆ABC$ is:

• A. $3:4:5$
• B. $5:12:13$
• C. $4:5:6$
• D. $5:6:7$

Answer 1

The correct option is C

$4:5:6$

Explanation of correct option.

Let $a,b,c$ in A.P.,

Therefore, it is given that $\angle C=2\angle A$

$$\begin{gathered} \Rightarrow \sin C=\sin 2A \\ \Rightarrow \sin C=2\sin A·\cos A \end{gathered}$$

$\frac{\sin C}{\sin A}=2\cos A$

$\Rightarrow \frac{\mathrm{c}}{\mathrm{a}}=2\frac{{\mathrm{b}}^2+{\mathrm{c}}^2-{\mathrm{a}}^2}{2\mathrm{bc}}$

Now, put $\mathrm{a}=\mathrm{b}-\lambda ,\mathrm{c}=\mathrm{b}+\lambda ,\lambda >0$

$\Rightarrow \lambda =\frac{\mathrm{b}}{5}$

$\Rightarrow b=5\lambda$

$\Rightarrow a=b-\frac{b}{5}=\frac{4}{5}b,c=b+\frac{b}{5}=\frac{6b}{5}$

Therefore,

$$\begin{gathered} a=4\lambda \\ b=5\lambda \\ c=6\lambda \end{gathered}$$

Thus the ratio required is $4:5:6$

Hence, Option (C) is correct.