Question

Let $AD,BE$ and $CF$ be the perpendiculars from the angular points of a $\Delta ABC$ upon the opposite sides.

What is the ratio of the perimeters of the $\Delta DEF$ and $\Delta ABC$

(where $r$ is the inradius and $R$ is circum-radius of the $\Delta ABC$)

• A. $\frac{2r}{R}$
• B. $\frac{r}{2R}$
• C. $\frac{r}{R}$
• D. $\frac{r}{3R}$

Answer 1

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

Perimeter of Pedal Triangle DEF = $4R\sin A\sin B\sin C$

Perimeter of Triangle ABC = $a+b+c$

$\frac{△DEF}{△ABC}=\frac{R.4\sin A\sin B\sin C}{(a+b+c)}\Rightarrow \frac{R.4\sin A\sin B\sin C}{2s}[s=\frac{a+b+c}{2}]=\frac{R.4\times \frac{a}{2R}\times \frac{b}{2R}\times \frac{c}{2R}}{2s}[R=\frac{a}{2\sin A}=\frac{b}{2\sin B}=\frac{c}{2\sin C}=\frac{abc}{4△}]$

$=4R\times \frac{abc}{2\times {8R}^3\times s}=\frac{abc}{{4R}^{2}S}\Rightarrow \frac{abc\times 4△}{4\times R\times abc\times s}\Rightarrow \frac{△}{Rs}\Rightarrow \frac{r}{R}$

Hence, $\frac{r}{R}$ is the correct answer.