Question

If perimeter of $△LMN$ and $△ABC$ are $\lambda$ and $\mu$, the value of $\frac{\lambda }{\mu }$ is

• A. $\frac{r}{R}$
• B. $\frac{R}{r}$
• C. $\frac{rR}{\Delta }$
• D. $\frac{\Delta }{rR}$

Answer 1

The correct option is A $\frac{r}{R}$

LMN is a pedal $△$, with sides $NM=a\cos A,NL=b\cos B,$ and $LM=c\cos C$

$\Rightarrow \lambda ($ perimeter of $LMN)=a\cos A+b\cos B+c\cos C$

$=\sum 2R\sin A\cos A=R\sum \sin 2A=R\left(4\sin A\sin B\sin C\right)$

$=4R\left(\frac{a}{2R}\right)\left(\frac{b}{2R}\right)\left(\frac{c}{2R}\right)=\frac{abc}{2R^2}=\frac{4R\Delta }{2R^2}(\because \Delta =\frac{abc}{4R})$

$=\frac{2\Delta }{R}=\frac{2\left(rs\right)}{R}=\frac{r}{R}\left(2s\right)\Rightarrow \lambda =\frac{r}{R}\left(\mu \right)\Rightarrow \frac{\lambda }{\mu }=\frac{r}{R}$