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Pedal Triangle

If x,y,z are ...

Question

If $x, y, z$ are the sides of pedal triangle of $△ A B C,$ then $x + y + z$ is equal to
(For $△ A B C,$ usual notations are used)

4 R sin A \cdot sin B \cdot sin C

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4 R sin 2 A \cdot sin 2 B \cdot sin 2 C

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\frac{Δ}{R}

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\frac{2 Δ}{R}

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Solution

The correct option is D $\frac{2 Δ}{R}$
We know, for a $△ A B C;$ sides of its pedal triangle are
$R sin 2 A, R sin 2 B, R sin 2 C$
$\Rightarrow x + y + z = R (sin 2 A + sin 2 B + sin 2 C) = 4 R sin A sin B sin C [∵ for △ A B C sin 2 A + sin 2 B + sin 2 C = 4 sin A sin B sin C] = 4 R \cdot \frac{a b c}{8 R^{3}} = \frac{a b c}{2 R^{2}} = \frac{2 Δ}{R} [∵ Δ = \frac{a b c}{4 R}]$

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