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in Trigonometry by (36.2k points)
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The angle of elevation of the top two vertical towers as seen from the middle point joining the feet of the towers are 60° and 30° respectively. The ratio of the heights of the towers is

(a) 2 : 1

(b) \(\sqrt3\) :1

(c) 3 : 2

(d) 3 : 1

1 Answer

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Best answer

(d) 3 : 1

Let PQ and RS be the two given tower and T, the middle point of the line QS, joining the feet of the towers.

Given,

∠PTQ = 60°,∠RTS = 30°,QT = TS.

In rt. Δ PQT,

tan 60° = \(\frac{PQ}{QT}\) 

⇒ \(\sqrt3\) = \(\frac{PQ}{QT}\)

⇒ QT = \(\frac{PQ}{\sqrt3}\) ...(i)

In rt. Δ RTS,

tan 30° = \(\frac{RS}{ST}\) 

⇒ \(\frac{1}{\sqrt3}=\frac{RS}{ST}\)

⇒ ST = RS\(\sqrt3\) ...(ii)

From (i) and (ii),

QT = ST

⇒ \(\frac{PQ}{\sqrt3}=RS\sqrt3\) 

⇒ \(\frac{PQ}{RS}=\frac{\sqrt3\times\sqrt3}{1}\)

⇒ PQ : RS = 3 : 1

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