Question

If $p_1,p_2,p_3$ are respectively the perpendiculars from the vertices of a triangle to the opposite sides, then $p_1{p}_2{p}_3=$

• A. $\frac{a^2{b}^2{c}^2}{R^2}$
• B. $\frac{a^2{b}^2{c}^2}{{4R}^2}$
• C. $\frac{4a^2{b}^2{c}^2}{R^2}$
• D. $\frac{a^2{b}^2{c}^2}{{8R}^2}$

Answer 1

The correct option is D $\frac{a^2{b}^2{c}^2}{{8R}^2}$

We have $\Delta =\frac{1}{2}a{p}_1=\frac{1}{2}b{p}_2=\frac{1}{2}c{p}_3$

$\therefore p_1=\frac{2\Delta }{a},p_2=\frac{2\Delta }{b},p_3=\frac{2\Delta }{c}$

$\therefore p_1{p}_2{p}_3=\frac{8{\Delta }^3}{abc}=\frac{8}{abc}{\left(\frac{abc}{4R}\right)}^3=\frac{a^2{b}^2{c}^2}{8R^2}$