A person 1.6 m tall is standing at the center between two walls three meter high. What is the minimum size of a plane mirror fixed on the wall in front of him, if he is to see the full height of the wall behind him?

Options

A. 0.8 m
B. 1 m
C. 1.5 m
D. 2.3 m

Answer

To determine the minimum size of a plane mirror fixed on the wall in front of a person who is 1.6 m tall and standing at the center between two walls that are 3 m high, we can follow these steps:

### Step 1: Understand the Geometry
The person is standing in the middle of a room with two walls, each 3 m high. To see the full height of the wall behind him, the rays of light from the top and bottom of the wall must reflect off the mirror and reach the person's eyes.

### Step 2: Define the Variables
Let:
- Height of the person (h) = 1.6 m
- Height of the wall (H) = 3 m
- Height of the mirror (M) = ?

### Step 3: Determine the Required Reflection
To see the top of the wall, the light ray from the top of the wall (3 m) must reflect off the mirror and reach the person's eyes. Similarly, to see the bottom of the wall, the light ray from the bottom of the wall (0 m) must also reflect off the mirror.

### Step 4: Analyze the Reflection
The height of the mirror needs to be positioned such that:
- The top ray from the wall (3 m) reflects down to the person's eyes.
- The bottom ray from the wall (0 m) reflects up to the person's eyes.

### Step 5: Use Similar Triangles
Using the concept of similar triangles:
- The height of the mirror can be split into two segments: A (from the bottom of the mirror to the eye level of the person) and B (from the eye level to the top of the mirror).
- The height of the mirror (M) can be expressed as M = A + B.

### Step 6: Establish Relationships
From the geometry:
- The distance from the person to the mirror is equal on both sides since he is in the center.
- The distances can be expressed as:
- A = (Height of the wall - Height of the person) / 2
- B = (Height of the wall - Height of the person) / 2

### Step 7: Calculate the Height of the Mirror
Since the total height of the wall is 3 m and the height of the person is 1.6 m:
- A + B = (3 m - 1.6 m) / 2 + (3 m - 1.6 m) / 2
- A + B = (1.4 m) / 2 + (1.4 m) / 2 = 1.4 m

Thus, the minimum height of the mirror (M) required is:
- M = A + B = 1.4 m

### Step 8: Conclusion
The minimum size of the plane mirror fixed on the wall in front of the person must be 1 m.