Properties of Triangles 2 Question 16
16. If $p _1, p _2, p _3$ are the perpendiculars from the vertices of a triangle to the opposite sides, then prove that
$$ p _1 p _2 p _3=\frac{a^{2} b^{2} c^{2}}{8 R^{3}} $$
(1978, 3M)
Integer Answer Type Question
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Solution:
- We know that, $\Delta=\frac{1}{2} a p _1$
$\Rightarrow \quad p _1=\frac{2 \Delta}{a}$
Similarly, $\quad p _2=\frac{2 \Delta}{b}$ and $p _3=\frac{2 \Delta}{c}$
Now,
$$ p _1 p _2 p _3=\frac{8 \Delta^{3}}{a b c} $$
Since, $\Delta=\frac{a b c}{4 R}$
$$ \therefore \quad p _1 p _2 p _3=\frac{8}{a b c} \cdot \frac{(a b c)^{3}}{64 R^{3}}=\frac{(a b c)^{2}}{8 R^{3}} $$
