Find the length of a chord which is at a distance of3 cm from the centre of a circle of radius 5 cm.

Answer

To find the length of a chord that is at a distance of 3 cm from the center of a circle with a radius of 5 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Geometry**:
- We have a circle with a radius (r) of 5 cm.
- The distance from the center of the circle to the chord (d) is 3 cm.

2. **Draw a Diagram**:
- Draw a circle and mark the center as O.
- Draw a chord AB such that the perpendicular distance from O to the chord AB is 3 cm. Let the point where the perpendicular meets the chord be M.

3. **Apply the Pythagorean Theorem**:
- In the right triangle OMA, where O is the center, M is the midpoint of the chord AB, and A is one endpoint of the chord:
- OM = d = 3 cm (distance from the center to the chord)
- OA = r = 5 cm (radius of the circle)
- AM = x (half the length of the chord)

4. **Set Up the Equation**:
- According to the Pythagorean theorem:
\[ OA^2 = OM^2 + AM^2 \]
- Substituting the known values:
\[ 5^2 = 3^2 + x^2 \]
- This simplifies to:
\[ 25 = 9 + x^2 \]

5. **Solve for x**:
- Rearranging the equation gives:
\[ x^2 = 25 - 9 \]
\[ x^2 = 16 \]
- Taking the square root of both sides:
\[ x = 4 \text{ cm} \]

6. **Calculate the Length of the Chord**:
- Since x is half the length of the chord, the full length of the chord AB is:
\[ AB = 2x = 2 \times 4 = 8 \text{ cm} \]

### Final Answer:
The length of the chord is **8 cm**.