Question

If area of $△LMN$ is $\Delta$', then the value of $\frac{{\Delta }^{\prime }}{\Delta }$ is

• A. $2\sin A\sin B\sin C$
• B. $2\cos A\cos B\cos C$
• C. $\sin A\sin B\sin C$
• D. $\cos A\cos B\cos C$

Answer 1

The correct option is B $2\cos A\cos B\cos C$

Area of pedal $△LMN$ is given by

${\Delta }^{\prime }=\frac{1}{2}{R}^2\sin 2A\sin 2B\sin 2C$

$=\frac{1}{2}{R}^2\left(8\sin A\cos A\sin B\cos B\sin C\cos C\right)$

$=2\left[2R^2\sin A\sin B\sin C\right]\cos A\cos B\cos C=2\Delta \cos A\cos B\cos C$

$\Rightarrow \frac{{\Delta }^{\prime }}{\Delta }=2\cos A\cos B\cos C$