.A house of height 100 m substends a right angle at the window of an opposite house. If the height of the window is 64 m, then the distance between the two houses is

Options

A. 48 m
B. 36 m
C. 54 m
D. 72 m

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the Problem
We have two houses. The first house (House A) has a height of 100 m, and the second house has a window at a height of 64 m. The line of sight from the top of House A to the window of House B forms a right angle.

### Step 2: Define the Variables
Let:
- \( A \) be the top of House A.
- \( B \) be the bottom of House A.
- \( C \) be the top of the window on House B.
- \( D \) be the bottom of House B (ground level).
- \( E \) be the point on the ground directly below point A.

### Step 3: Calculate the Height Difference
The height of House A is 100 m, and the height of the window (C) is 64 m. The height difference between point A and point C is:
\[ AE = AB - CD = 100 - 64 = 36 \text{ m} \]

### Step 4: Set Up the Right Triangle
In triangle \( AEC \):
- The height \( AE = 36 \) m (the vertical distance between the top of House A and the window).
- The distance \( EC \) is the horizontal distance between the two houses.

### Step 5: Use the Tangent Function
Since angle \( \theta \) is formed at point C, we can use the tangent function:
\[ \tan(\theta) = \frac{AE}{EC} \]
Thus,
\[ \tan(\theta) = \frac{36}{EC} \]

### Step 6: Set Up Another Right Triangle
In triangle \( BDC \):
- The height from the window to the ground is \( 64 \) m.
- The horizontal distance \( EC \) is the same as \( BD \).

Using the tangent function again:
\[ \tan(\theta) = \frac{64}{EC} \]

### Step 7: Equate the Two Tangent Expressions
Since both expressions equal \( \tan(\theta) \):
\[ \frac{36}{EC} = \frac{64}{EC} \]

### Step 8: Cross Multiply to Solve for EC
Cross multiplying gives:
\[ 36 \cdot EC = 64 \cdot EC \]
This simplifies to:
\[ 36 \cdot EC = 64 \cdot EC \]

### Step 9: Solve for EC
Rearranging gives:
\[ EC^2 = 36 \cdot 64 \]
Calculating \( 36 \cdot 64 \):
\[ EC^2 = 2304 \]
Taking the square root:
\[ EC = \sqrt{2304} = 48 \text{ m} \]

### Final Answer
The distance between the two houses is \( 48 \) meters.

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