An optician while testing the eyes of a patient keeps a charge of letters 3 m behind the patient and asks him to see the letters on the image of chart formed in plane mirror kept at distance 2 m in front of him. At what distance is the chart seen by the patient?

Answer

To solve the problem, we need to determine the distance at which the patient sees the chart using the properties of reflection in a plane mirror.

### Step-by-Step Solution:

1. **Identify the Positions**:
- The patient is sitting at a certain position.
- The chart of letters is placed 3 meters behind the patient.

2. **Determine the Distance of the Chart from the Patient**:
- Since the chart is 3 meters behind the patient, the distance from the patient to the chart is 3 meters.

3. **Locate the Mirror**:
- The mirror is placed 2 meters in front of the patient.

4. **Calculate the Total Distance from the Chart to the Mirror**:
- The total distance from the chart (behind the patient) to the mirror (in front of the patient) is:
\[ \text{Distance from Patient to Chart} + \text{Distance from Patient to Mirror} = 3 \text{ m} + 2 \text{ m} = 5 \text{ m} \]

5. **Determine the Image Distance**:
- In a plane mirror, the image of an object appears at the same distance behind the mirror as the object is in front of it. Therefore, the image of the chart will be 5 meters behind the mirror.

6. **Calculate the Distance from the Patient to the Image**:
- The distance from the patient to the mirror is 2 meters, and the image of the chart is located 5 meters behind the mirror. Thus, the total distance from the patient to the image of the chart is:
\[ \text{Distance from Patient to Mirror} + \text{Distance from Mirror to Image} = 2 \text{ m} + 5 \text{ m} = 7 \text{ m} \]

7. **Final Answer**:
- Therefore, the distance at which the chart is seen by the patient is **7 meters**.