A house subtends a right angle at the window of an opposite house and the angle of elevation of the window for the bottom of the first house is `60^@` If the distance between the two houses be `6m` then the height of the first house is

Options

A. `8sqrt3`m
B. `6sqrt3`m
C. `4sqrt3`m
D. none of these

Explanation

To solve the problem step by step, let's break it down:

### Step 1: Understand the Problem
We have two houses, and the first house subtends a right angle at the window of the opposite house. The angle of elevation from the bottom of the first house to the window of the second house is \(60^\circ\). The distance between the two houses is \(6 \, m\). We need to find the height of the first house.

### Step 2: Draw a Diagram
Let's label the points:
- Let \(A\) be the bottom of the first house.
- Let \(B\) be the top of the first house (height \(h_1\)).
- Let \(C\) be the bottom of the second house.
- Let \(D\) be the window of the second house (height \(h_2\)).
- The distance \(AC = 6 \, m\).

### Step 3: Identify the Angles
From point \(A\), the angle of elevation to point \(D\) is \(60^\circ\). Since the house subtends a right angle at the window, the angle at point \(C\) is \(90^\circ\).

### Step 4: Use Trigonometry
In triangle \(ABD\):
- The angle \(BAD = 60^\circ\).
- The opposite side is \(h_2\) (height of the window), and the adjacent side is \(AC = 6 \, m\).

Using the tangent function:
\[ \tan(60^\circ) = \frac{h_2}{6} \]
Since \(\tan(60^\circ) = \sqrt{3}\):
\[ \sqrt{3} = \frac{h_2}{6} \]
Thus,
\[ h_2 = 6\sqrt{3} \]

### Step 5: Analyze Triangle \(ACD\)
In triangle \(ACD\):
- The angle \(CAD = 30^\circ\) (since \(90^\circ - 60^\circ = 30^\circ\)).
- The opposite side is \(h_1 - h_2\) (the height of the first house minus the height of the window), and the adjacent side is \(AC = 6 \, m\).

Using the tangent function again:
\[ \tan(30^\circ) = \frac{h_1 - h_2}{6} \]
Since \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\):
\[ \frac{1}{\sqrt{3}} = \frac{h_1 - h_2}{6} \]
Thus,
\[ h_1 - h_2 = \frac{6}{\sqrt{3}} = 2\sqrt{3} \]

### Step 6: Substitute \(h_2\)
Now we substitute \(h_2 = 6\sqrt{3}\) into the equation:
\[ h_1 - 6\sqrt{3} = 2\sqrt{3} \]
Solving for \(h_1\):
\[ h_1 = 2\sqrt{3} + 6\sqrt{3} = 8\sqrt{3} \]

### Step 7: Final Answer
The height of the first house is:
\[ h_1 = 8\sqrt{3} \, m \]

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