Question

If areas of $△$'s $AMN,BNL$ and $CLM$ are ${\Delta }_1,{\Delta }_2$ and ${\Delta }_3$ respectively, then the value of ${\Delta }_1+{\Delta }_2+{\Delta }_3$ is

• A. $\Delta (2+2\cos A\cos B\cos C)$
• B. $\Delta (2+2\sin A\sin B\sin C)$
• C. $\Delta (1-2\cos A\cos B\cos C)$
• D. $\Delta (1-2\sin A\sin B\sin C)$

Answer 1

The correct option is C $\Delta (1-2\cos A\cos B\cos C)$

${\Delta }_1+{\Delta }_2+{\Delta }_3=\Delta -($ area of pedal $△LMN)$

$=\Delta -\frac{1}{2}{R}^2\sin 2A\sin 2B\sin 2C$ ....(1)

Also $\Delta =\frac{1}{2}ab\sin C=\frac{1}{2}\left(2R\sin A\right)\left(2R\sin B\right)\left(2R\sin C\right)$

$\Rightarrow \Delta =2R^2\sin A\sin B\sin C$ ...(2)

From (1), we get ${\Delta }_1+{\Delta }_2+{\Delta }_3=\Delta -2\left(2R^2\sin A\sin B\sin C\right)\times \cos A\cos B\cos C$

$=\Delta -2\Delta \cos A\cos B\cos C$ (From (2))

$=\Delta (1-2\cos A\cos B\cos C)$