Length of a chord `PQ` of a circle with centre O is 4 cm. If the distance of `PQ` from the centre of circle is 2 cm, then the length of the diameter is:

Options

A. `2sqrt(2) cm.`
B. `3sqrt(2) cm`.
C. `5sqrt(2) cm`.
D. `4sqrt(2) cm`.

Answer

To find the diameter of the circle given the length of the chord \( PQ \) and its distance from the center \( O \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Information**:
- Length of chord \( PQ = 4 \) cm
- Distance from the center \( O \) to the chord \( PQ = 2 \) cm

2. **Draw the Circle**:
- Draw a circle with center \( O \).
- Mark the chord \( PQ \) such that the distance from \( O \) to \( PQ \) is perpendicular. Let the point where the perpendicular from \( O \) meets \( PQ \) be \( M \).

3. **Divide the Chord**:
- Since \( M \) is the midpoint of the chord \( PQ \) (as the perpendicular from the center to the chord bisects the chord), we can say:
\[ PM = MQ = \frac{PQ}{2} = \frac{4}{2} = 2 \text{ cm} \]

4. **Apply the Pythagorean Theorem**:
- In the right triangle \( OMQ \):
- \( OM = 2 \) cm (distance from center to chord)
- \( MQ = 2 \) cm (half of the chord)
- Let \( OQ \) be the radius \( r \) of the circle.
- According to the Pythagorean theorem:
\[ OQ^2 = OM^2 + MQ^2 \]
Substituting the known values:
\[ OQ^2 = 2^2 + 2^2 = 4 + 4 = 8 \]
Therefore:
\[ OQ = \sqrt{8} = 2\sqrt{2} \text{ cm} \]

5. **Calculate the Diameter**:
- The diameter \( D \) of the circle is twice the radius:
\[ D = 2 \times OQ = 2 \times 2\sqrt{2} = 4\sqrt{2} \text{ cm} \]

### Final Answer:
The length of the diameter of the circle is \( 4\sqrt{2} \) cm.

---