# If the angles of elevation of the top of a  tower from three collinear points A, B and C, on a line leading to the foot of thetower, are 300, 450 and 600 respectively,then the ratio, AB : BC, is : Option 1) Option 2) Option 3) Option 4)

As we learnt in

Height and Distances -

The height or length of an object or the distance between two distant objects can be determined with the help of trigonometric ratios.

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In  $\Delta PQA, \tan 30^{o}=\frac{PQ}{AQ}$

$\Rightarrow AQ=\sqrt3h$                                ...................... ( 1 )

In $\Delta PQB, \tan 45^{o}=\frac{PQ}{BQ}$

$\Rightarrow BQ=h$                                         .................... ( 2 )

In $\Delta PQC, \tan 60^{o}=\frac{PQ}{CQ}$

$\Rightarrow CQ=\frac{h}{\sqrt 3}$                                    ..................... ( 3 )

Now,  $\frac{AB}{BC}=\frac{AQ-BQ}{BQ-CQ}=\frac{\sqrt3-1}{1-\frac{1}{\sqrt 3}}$

$=\sqrt 3\:\: \frac{(\sqrt 3-1)}{\sqrt 3-1}$

$=\sqrt 3$

$AB:BC=\sqrt 3:1$

Option 1)

This option is correct.

Option 2)

This option is incorrect.

Option 3)

This option is incorrect.

Option 4)

This option is incorrect.

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