First slide

Applications of trigonometry

Question

$A tower A B C D stands on a level ground with foot A . At a point P on the ground, the$ $portion A B, A C and A D subtends angles α, β and γ respectively. If$ $A B = a, A C = b, A D = c, A P = x and α + β + γ = 180 °, then (a + b + c) x^{2} =$

Moderate

A

abc
B

$\frac{1}{a b c}$
C

a + b - c
D

a - b - c

Solution

$From, Δ P A B, \tan α = \frac{a}{x}$
$From, Δ P A C, \tan β = \frac{b}{x}$
$From, Δ P A D, \tan γ = \frac{c}{x}$
$Given α + β + γ = 180$
$\Rightarrow \tan α + \tan β + \tan γ = \tan α \tan β \tan γ$
$\Rightarrow \frac{a}{x} + \frac{b}{x} + \frac{c}{x} = \frac{a b c}{x^{3}}$
$\Rightarrow (a + b + c) x^{2} = a b c$

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