Two circles of radii 10 cm and 8 cm intersect each other, and the length of the common chord is 12 cm. Find the distance between their centre.

Explanation

To find the distance between the centers of two intersecting circles with given radii and the length of the common chord, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Radius of Circle 1 (O) = 10 cm
- Radius of Circle 2 (O') = 8 cm
- Length of the common chord (AB) = 12 cm

2. **Calculate Half the Length of the Common Chord**:
- Since the common chord AB is 12 cm, the distance from the center of the circles to the midpoint of the chord (AD) is half of AB.
- \( AD = \frac{AB}{2} = \frac{12}{2} = 6 \) cm

3. **Use the Right Triangle Formed by the Radius and the Chord**:
- For Circle 1 (O):
- In triangle OAD, we can apply the Pythagorean theorem:
\[ OA^2 = OD^2 + AD^2 \]
- Here, \( OA = 10 \) cm and \( AD = 6 \) cm.
- Therefore,
\[ OD^2 = OA^2 - AD^2 = 10^2 - 6^2 = 100 - 36 = 64 \]
- Thus, \( OD = \sqrt{64} = 8 \) cm.

4. **Repeat for Circle 2 (O')**:
- For Circle 2 (O'):
- In triangle O'DA, we again apply the Pythagorean theorem:
\[ O'A^2 = O'D^2 + AD^2 \]
- Here, \( O'A = 8 \) cm and \( AD = 6 \) cm.
- Therefore,
\[ O'D^2 = O'A^2 - AD^2 = 8^2 - 6^2 = 64 - 36 = 28 \]
- Thus, \( O'D = \sqrt{28} = 2\sqrt{7} \) cm.

5. **Calculate the Distance Between the Centers (OO')**:
- The total distance between the centers O and O' is the sum of OD and O'D:
\[ OO' = OD + O'D = 8 + 2\sqrt{7} \text{ cm} \]

### Final Answer:
The distance between the centers of the two circles is \( 8 + 2\sqrt{7} \) cm.

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