A carrom board `(4ftxx4ft )` has the queen at the centre. The queen hit by the striker moves to the front edge, rebounds and goes in the hole behind the striking line. Find the magnitude of displacement of the queen a. from the centre to the front edge b. from the front edge to the hole and c. from the centre to the hole.

Explanation

To solve the problem step by step, we will find the displacement of the queen in three parts: from the center to the front edge, from the front edge to the hole, and from the center to the hole.

### Step 1: Displacement from the center to the front edge (O to A)

1. **Identify the dimensions**: The carrom board is 4 ft x 4 ft, so the center (O) is at (2 ft, 2 ft).
2. **Determine the front edge**: The front edge is at (2 ft, 0 ft).
3. **Calculate the displacement**: The displacement OA can be calculated using the Pythagorean theorem:
\[ OA = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
where \( (x_1, y_1) = (2, 2) \) and \( (x_2, y_2) = (2, 0) \).
\[ OA = \sqrt{(2 - 2)^2 + (0 - 2)^2} = \sqrt{0 + 4} = 2 \text{ ft} \]

### Step 2: Displacement from the front edge to the hole (A to B)

1. **Identify the hole position**: The hole is behind the striking line, at (2 ft, -2 ft).
2. **Calculate the displacement AB**: The displacement AB can also be calculated using the Pythagorean theorem:
\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
where \( (x_1, y_1) = (2, 0) \) and \( (x_2, y_2) = (2, -2) \).
\[ AB = \sqrt{(2 - 2)^2 + (-2 - 0)^2} = \sqrt{0 + 4} = 2 \text{ ft} \]

### Step 3: Displacement from the center to the hole (O to B)

1. **Calculate the displacement OB**: The displacement OB can be calculated using the Pythagorean theorem:
\[ OB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
where \( (x_1, y_1) = (2, 2) \) and \( (x_2, y_2) = (2, -2) \).
\[ OB = \sqrt{(2 - 2)^2 + (-2 - 2)^2} = \sqrt{0 + 16} = 4 \text{ ft} \]

### Summary of Results

- **Displacement from center to front edge (O to A)**: 2 ft
- **Displacement from front edge to hole (A to B)**: 2 ft
- **Displacement from center to hole (O to B)**: 4 ft