Two rays are drawn through a point $A$ at an angle of $30^{\circ}$ . A point $B$ is taken on one of them at distance a from the point $A$ . A perpendicular is drawn from the point $B$ to the other ray, another perpendicular is drawn from its foot to $A B$ , and so on, then the length of the resulting infinite polygonal line is

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$a (2 + \sqrt{3})$

$a (2 - \sqrt{3})$

None of these

$2 \sqrt{3} a$

answer is B.

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Detailed Solution

$A B = a$
$B B_{1} = a / 2$
$B_{2} B = a / 4$
$△ A B_{2} B_{3} \sim △ A B B_{1}$

$\frac{B_{2} B_{3}}{B_{3} B_{1}} = \frac{3}{4} = \frac{A B_{3}}{A B_{1}}$

infinite GP series with $r = 3 / 4$

$S_{\infty} = \frac{a}{1 - r} \Rightarrow$ $\frac{\frac{a}{4} (\sqrt{3 + \sqrt{4}})}{1 - 3 / 4}$ $= a (2 + \sqrt{3})$
$S_{\infty} = a (2 + \sqrt{3})$

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