Question

A monument ABCD stands on level ground. At a point P on ground AB,AC,AD substend angles $\alpha$, $\beta$, $\gamma$ respectively . if AB = a , AC = b , AD = c , AP = x and
$\alpha + \beta + \gamma = 180$
Then x^2 is equal to​

Answer 1

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$a \: = \: AB = AP \tan( \alpha ) = x \tan( \alpha )$

$b \: = \: AC = x \tan( \beta )$

$c \: = AD = x \tan( \gamma )$

so that

$\tt\:a + b + c \: = \: x( \tan \alpha + \tan\beta + \tan \gamma ) \\ \\ = \: x \tan \alpha \tan\beta \tan \gamma$

$\tt\:abc \: = \: x {}^{3} \tan \alpha \tan \beta \tan \gamma$

$\tt{\boxed {x {}^{2} \: = \: \dfrac{abc}{a + b + c}}}$

Answer 2

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