Question

If two towers of heights h₁ and h₂ subtend angles of 60º and 30º respectively at the mid-point of line segment joining their feet, then the ratio of their heights h₁ : h₂ is

• A. 1 : 2
• B. 2 : 1
• C. 1 : 3
• D. 3 : 1

Answer 1

The Correct Option is D

Given: 

Two towers of heights \( h_1 \) and \( h_2 \) subtend angles of \( 60^\circ \) and \( 30^\circ \) respectively at the midpoint of the line segment joining their feet.

Step 1: Define the Right Triangles

Let the distance from the midpoint to the base of each tower be \( d \). Then, using the tangent function: \[ \tan 60^\circ = \frac{h_1}{d}, \quad \tan 30^\circ = \frac{h_2}{d} \]

Step 2: Solve for Heights

We know: \[ \tan 60^\circ = \sqrt{3}, \quad \tan 30^\circ = \frac{1}{\sqrt{3}} \] So, \[ h_1 = d\sqrt{3}, \quad h_2 = \frac{d}{\sqrt{3}} \]

Step 3: Compute the Ratio

\[ \frac{h_1}{h_2} = \frac{d\sqrt{3}}{\frac{d}{\sqrt{3}}} = 3 \] \[ h_1 : h_2 = 3 : 1 \]

Final Answer: 3:1

Answer 2

The problem requires finding the ratio of the heights of two towers based on the angles they subtend at the midpoint of the line segment joining their feet. 
Let's denote the midpoint as M and the towers as A and B with heights h₁ and h₂, subtending angles 60º and 30º respectively at M.

From point M, if we drop perpendiculars to the lines of the towers, by trigonometry, we have:

For tower A:
tan 60º = ^h₁/_x
And for tower B:
tan 30º = ^h₂/_x
where x is the horizontal distance from M to the base of each tower.
We know:
tan 60º = √3
tan 30º = ¹/_√3
Now, substituting the values:
√3 = ^h₁/_x  (1)
¹/_√3 = ^h₂/_x  (2)
From equation (1), h₁ = √3x
From equation (2), h₂ = x/√3
Taking the ratio:
^h₁/_h2 = ^√3x/_(x/√3)
This simplifies to:
(√3x) × (√3/x) = 3
Therefore, the ratio h₁ : h₂ is 3 : 1.