If two towers of height h₁ and h₂ subtend angles of 60° and 30° respectively at the midpoint of the line joining their feet, then show that h₁ : h₂ = 3 : 1.

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Solution

Let $A B$ and $C D$ be the two towers subtending angles $60^{o} and 30^{o}$ , respectively.
Now,
∠ $A P B = 30^{o}$ and ∠ $D P C = 60^{o}$
Let $P$ be the midpoint of the line segment joining the two towers.
Let:
$A B = h_{1}$ , $C D = h_{2}$ and $B P = C P = x$

From ∆ $D C P$ , we have:
$\frac{D C}{P C} = \tan 30^{o} = \frac{1}{\sqrt{3}}$
⇒ $\frac{h_{2}}{x} = \frac{1}{\sqrt{3}}$
⇒ $x = h_{2} \sqrt{3}$ ...(i)

From ∆ $A B P$ , we have:
$\frac{A B}{B P} = \tan 60^{o} = \sqrt{3}$

⇒ $\frac{h_{1}}{x} = \sqrt{3}$
⇒ $x = \frac{h_{1}}{\sqrt{3}}$ ...(ii)
From (i) and (ii), we get:
$\frac{h_{1}}{\sqrt{3}} = h_{2} \sqrt{3}$

⇒ $\frac{h_{1}}{h_{2}} = \frac{3}{1}$
∴ $h_{1} : h_{2} = 3 : 1$

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If two towers of height $h_{1} a n d h_{2}$ subtend angles $45^{\circ} a n d 30^{\circ}$ respectively at the midpoint of the line joining their feet, then find the ratio of

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Q. Two towers of heights

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subtend angles

60^{0}

and

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respectively at the midpoint of the line joining their feet.

h_{1} : h_{2} =

If two towers of height $h_{1}$ and $h_{2}$ subtend angles $45^{\circ}$ and $30^{\circ}$ respectively at the midpoint of the line joining their feet, then find the ratio of $h_{1} : h_{2}$ .

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and

h_{2}

substends angles of

60^{\circ}

and

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mid point of the line joining their feet. Then what is

h_{1} : h_{2}

Q. Assertion (A)
If two towers of height h₁ and h₂ subtend angles of 60° and 30° respectively at the midpoint of the line joining their feet, then h₁ : h₂ = 3 : 1.

Reason (R)
Figure
In the given figure, we have:

\frac{h_{1}}{x} = \tan 60 ° and \frac{h_{2}}{x} = \tan 30 °

(a) Both Assertion (A) and Reason (R) are true and (R) is a correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).
(c) Assertion (A) is true and Reason (R) is false.
(d) Assertion (A) is false and Reason (R) is true.

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If two towers of height h1 and h2 subtend angles of 60° and 30° respectively at the midpoint of the line joining their feet, then show that h1 : h2 = 3 : 1.

If two towers of height h₁ and h₂ subtend angles of 60° and 30° respectively at the midpoint of the line joining their feet, then show that h₁ : h₂ = 3 : 1.