Find the minimum size of mirror required to see the full image of a wall behind a man standing at the centre of room, where H is the height of wall.

Answer

To find the minimum size of a mirror required for a man standing at the center of a room to see the full image of a wall behind him, we can follow these steps:

### Step 1: Understand the Geometry
- The man is standing at the center of the room, and there is a wall behind him with a height \( H \).
- The distance from the man to the wall is \( d \), and since he is at the center, the distance from him to the wall is \( y/2 \) where \( y \) is the total distance from the wall to the mirror.

### Step 2: Set Up the Triangle
- When the man looks at the wall, the light from the top of the wall reflects off the mirror and reaches his eyes.
- Let’s denote the height of the man’s eyes from the ground as \( x \).
- The angle of incidence is equal to the angle of reflection.

### Step 3: Establish Relationships
- The height of the wall \( H \) can be expressed in terms of the height \( x \) and the distance \( d \) from the mirror to the wall.
- The relationship can be established as:
\[ H = 2x + 2d \]

### Step 4: Express \( d \) in Terms of \( x \)
- From the geometry of the situation, we can derive that the distance \( d \) is related to \( x \) by:
\[ d = 2x \]

### Step 5: Substitute \( d \) into the Height Equation
- Substituting \( d = 2x \) into the height equation gives:
\[ H = 2x + 2(2x) = 2x + 4x = 6x \]

### Step 6: Solve for \( x \)
- Rearranging gives:
\[ x = \frac{H}{6} \]

### Step 7: Find the Minimum Size of the Mirror
- The minimum size of the mirror \( mn \) required to see the full image is:
\[ mn = 2x = 2 \left(\frac{H}{6}\right) = \frac{H}{3} \]

### Conclusion
- Therefore, the minimum size of the mirror required for the man to see the full image of the wall is:
\[ \text{Minimum size of mirror} = \frac{H}{3} \]