A ceiling fan having 4 blades has a rotational symmetry of the order `2 xx K.` Find K.

Answer

To find the value of \( K \) in the rotational symmetry of the ceiling fan with 4 blades, we can follow these steps:

### Step 1: Understand the concept of rotational symmetry
Rotational symmetry refers to the number of times a shape can be rotated around a central point and still look the same within one complete turn (360 degrees).

### Step 2: Identify the number of blades
The ceiling fan has 4 blades. We can label them as A, B, C, and D.

### Step 3: Determine the angles of rotation
Since there are 4 blades, we can find the angle of rotation for each symmetry position. The angle of rotation can be calculated as:
\[ \text{Angle of rotation} = \frac{360^\circ}{\text{Number of blades}} = \frac{360^\circ}{4} = 90^\circ \]

### Step 4: Identify the positions after rotation
- When the fan is rotated by \( 90^\circ \):
- Blade A moves to the position of Blade B
- Blade B moves to the position of Blade C
- Blade C moves to the position of Blade D
- Blade D moves to the position of Blade A

- When the fan is rotated by \( 180^\circ \):
- Blade A moves to the position of Blade C
- Blade B moves to the position of Blade D
- Blade C moves to the position of Blade A
- Blade D moves to the position of Blade B

- When the fan is rotated by \( 270^\circ \):
- Blade A moves to the position of Blade D
- Blade B moves to the position of Blade A
- Blade C moves to the position of Blade B
- Blade D moves to the position of Blade C

- When the fan is rotated by \( 360^\circ \):
- All blades return to their original positions.

### Step 5: Count the number of symmetries
From the above rotations, we can see that the fan looks the same after:
1. \( 90^\circ \)
2. \( 180^\circ \)
3. \( 270^\circ \)
4. \( 360^\circ \)

Thus, there are a total of 4 positions where the fan looks the same.

### Step 6: Relate the order of rotational symmetry to \( K \)
The order of rotational symmetry is given as \( 2 \times K \). Since we found that the order of rotational symmetry is 4, we can set up the equation:
\[ 2 \times K = 4 \]

### Step 7: Solve for \( K \)
To find \( K \), divide both sides of the equation by 2:
\[ K = \frac{4}{2} = 2 \]

### Final Answer
The value of \( K \) is \( 2 \).

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