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Medians of a Triangle and the Drawing Method

Let p, q an...

Question

Let $p, q$ and $r$ be the altitudes of a triangle with area $S$ and perimeter $2 t$ , then the value of $\frac{1}{p} + \frac{1}{q} + \frac{1}{r}$ is

\frac{S}{t}

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\frac{t}{S}

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\frac{S}{2 t}

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\frac{2 S}{t}

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Solution

The correct option is B $\frac{t}{S}$
According to question, $S = \frac{1}{2} a \cdot p$
$= \frac{1}{2} b \cdot q = \frac{1}{2} c \cdot r$

$∴ \frac{1}{p} = \frac{a}{2 S}, \frac{1}{q} = \frac{b}{2 S}, \frac{1}{r} = \frac{c}{2 S}$
$∴ \frac{1}{p} + \frac{1}{q} + \frac{1}{r} = \frac{1}{2 S} (a + b + c) = \frac{2 t}{2 S} = \frac{t}{S}$

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