In a circle with radius 5 cm, the length of a chord lying at distance 4 cm from the centre is ......... cm.

Options

A. 3
B. 6
C. 12
D. 15

Answer

To find the length of a chord in a circle with a given radius and distance from the center, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Radius of the circle (r) = 5 cm
- Distance from the center to the chord (d) = 4 cm

2. **Draw the Circle and Chord:**
- Draw a circle with center O and radius 5 cm.
- Draw a chord AB such that the perpendicular distance from the center O to the chord AB is 4 cm. Let the point where the perpendicular from O meets the chord be point P.

3. **Understand the Geometry:**
- In triangle OAP (where A is one endpoint of the chord and P is the foot of the perpendicular from O), we have:
- OP = 4 cm (the distance from the center to the chord)
- OA = 5 cm (the radius of the circle)
- AP = x (half the length of the chord)

4. **Apply the Pythagorean Theorem:**
- According to the Pythagorean theorem:
\[ OA^2 = OP^2 + AP^2 \]
- Substituting the known values:
\[ 5^2 = 4^2 + x^2 \]
- This simplifies to:
\[ 25 = 16 + x^2 \]

5. **Solve for x:**
- Rearranging the equation gives:
\[ x^2 = 25 - 16 \]
\[ x^2 = 9 \]
- Taking the square root:
\[ x = 3 \text{ cm} \]

6. **Find the Length of the Chord:**
- Since AP = PB (the perpendicular from the center bisects the chord), the total length of the chord AB is:
\[ AB = AP + PB = 2 \times AP = 2 \times 3 = 6 \text{ cm} \]

### Final Answer:
The length of the chord AB is **6 cm**.