A man is sitting in a room at 2m from a wall `W_1`, wants to see the full height of the wall `W_2` behind him 4m high and 6m away from the facing wall `W_1`. What is the minimum vertical length of mirror on the facing wall required for the purpose?

Options

A. 4m
B. 2m
C. 3m
D. 1m

Explanation

To solve the problem, we need to determine the minimum vertical length of a mirror on wall \( W_1 \) that allows the man to see the entire height of wall \( W_2 \). Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Setup
- The man is sitting 2 meters away from wall \( W_1 \).
- Wall \( W_2 \) is 6 meters away from wall \( W_1 \) and has a height of 4 meters.
- The total distance from the man to wall \( W_2 \) is \( 2 + 6 = 8 \) meters.

### Step 2: Visualize the Reflection
- The man needs to see the top of wall \( W_2 \) (4 meters high) in the mirror on wall \( W_1 \).
- For the man to see the top of wall \( W_2 \), the light rays from the top of wall \( W_2 \) must reflect off the mirror on wall \( W_1 \) and reach the man's eyes.

### Step 3: Determine the Geometry
- Let’s denote the height of the mirror as \( L \).
- The light ray from the top of wall \( W_2 \) will reflect off the mirror at a certain angle to reach the man.
- The distance from the man to wall \( W_2 \) is 8 meters, and the height of wall \( W_2 \) is 4 meters.

### Step 4: Use Similar Triangles
- We can set up a proportion using similar triangles.
- The triangle formed by the height of wall \( W_2 \) (4 meters) and the distance from the man to wall \( W_2 \) (8 meters) is similar to the triangle formed by the height of the mirror \( L \) and the distance from the man to wall \( W_1 \) (2 meters).

### Step 5: Set Up the Proportion
- From the similar triangles, we have:
\[ \frac{L}{4} = \frac{2}{8} \]

### Step 6: Solve for \( L \)
- Cross-multiplying gives:
\[ 8L = 4 \times 2 \]
\[ 8L = 8 \]
\[ L = 1 \text{ meter} \]

### Conclusion
The minimum vertical length of the mirror required on wall \( W_1 \) for the man to see the full height of wall \( W_2 \) is **1 meter**.