If two towers of height `h_(1)` and `h_(2)` subted angle of `60^(@)` and `30^(@)` respectively at the mid point of the line joining their feet, then find `h_(1):h_(2)`.

Answer

To find the ratio of the heights of the two towers \( h_1 \) and \( h_2 \) given the angles they subtend at the midpoint of the line joining their feet, we can follow these steps:

### Step 1: Understand the Setup
Let the heights of the two towers be \( h_1 \) and \( h_2 \). The angle subtended by the first tower (height \( h_1 \)) at the midpoint is \( 60^\circ \), and the angle subtended by the second tower (height \( h_2 \)) at the same point is \( 30^\circ \).

### Step 2: Define the Midpoint
Let the distance from the midpoint to the foot of each tower be \( x \). Therefore, the total distance between the feet of the two towers is \( 2x \).

### Step 3: Use Trigonometric Ratios
For the first tower (height \( h_1 \)):
- In triangle formed by the height \( h_1 \) and the distance \( x \):
\[ \tan(60^\circ) = \frac{h_1}{x} \]
Since \( \tan(60^\circ) = \sqrt{3} \), we can write:
\[ \sqrt{3} = \frac{h_1}{x} \implies h_1 = \sqrt{3}x \]

For the second tower (height \( h_2 \)):
- In triangle formed by the height \( h_2 \) and the distance \( x \):
\[ \tan(30^\circ) = \frac{h_2}{x} \]
Since \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \), we can write:
\[ \frac{1}{\sqrt{3}} = \frac{h_2}{x} \implies h_2 = \frac{x}{\sqrt{3}} \]

### Step 4: Find the Ratio \( h_1:h_2 \)
Now we have:
\[ h_1 = \sqrt{3}x \quad \text{and} \quad h_2 = \frac{x}{\sqrt{3}} \]
To find the ratio \( h_1:h_2 \):
\[ \frac{h_1}{h_2} = \frac{\sqrt{3}x}{\frac{x}{\sqrt{3}}} = \sqrt{3}x \cdot \frac{\sqrt{3}}{x} = 3 \]
Thus, the ratio \( h_1:h_2 = 3:1 \).

### Final Answer
The ratio of the heights of the two towers is \( h_1:h_2 = 3:1 \).

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