Read the statements carefully and select the correct option.
Statement-I : If two circles with centres A and B intersect each other at points M and N, then the line joining the centres AB bisects the common chord MN at right angle.
Statement-II : Two circles of radii 10 cm and 8 cm intersect each other and the length of common chord is 12 cm. Then the distance between their centres is 8 cm.

Options

A. Both Statement-I and Statement-II are true.
B. Both Statement-I and Statement-II are false.
C. Statement-I is false but Statement-II is true.
D. Statement-I is true but Statement-II is false.

Answer

To solve the given problem, we will analyze both statements step by step.

### Step 1: Analyze Statement I
**Statement-I:** If two circles with centres A and B intersect each other at points M and N, then the line joining the centres AB bisects the common chord MN at right angle.

1. **Understanding the Geometry**: When two circles intersect, they create a common chord MN. The centers of the circles are points A and B.
2. **Perpendicular Bisector Concept**: A fundamental property of circles states that the line segment joining the centers of two intersecting circles is perpendicular to the common chord and bisects it.
3. **Conclusion for Statement I**: Therefore, Statement I is **True**.

### Step 2: Analyze Statement II
**Statement-II:** Two circles of radii 10 cm and 8 cm intersect each other and the length of the common chord is 12 cm. Then the distance between their centres is 8 cm.

1. **Given Data**:
- Radius of Circle A (r1) = 10 cm
- Radius of Circle B (r2) = 8 cm
- Length of common chord MN = 12 cm

2. **Finding the Half Length of the Chord**:
- Half of the common chord (MN) = 12 cm / 2 = 6 cm.

3. **Using Right Triangle Properties**:
- The distance from the center of each circle to the midpoint of the chord (let's call it O) is perpendicular to the chord.
- Let AO be the distance from center A to the midpoint O of the chord MN.
- Let BO be the distance from center B to the midpoint O of the chord MN.

4. **Applying Pythagorean Theorem**:
- For circle A:
\[ AO^2 + OM^2 = r_1^2 \implies AO^2 + 6^2 = 10^2 \implies AO^2 + 36 = 100 \implies AO^2 = 64 \implies AO = 8 \text{ cm} \]
- For circle B:
\[ BO^2 + OM^2 = r_2^2 \implies BO^2 + 6^2 = 8^2 \implies BO^2 + 36 = 64 \implies BO^2 = 28 \implies BO = \sqrt{28} = 2\sqrt{7} \text{ cm} \]

5. **Finding the Distance Between Centers A and B**:
- The distance AB can be calculated using the formula:
\[ AB = AO + BO = 8 + 2\sqrt{7} \]
- Since \(2\sqrt{7} \approx 5.29\), we find that \(AB \approx 13.29\) cm, which is greater than 8 cm.

6. **Conclusion for Statement II**: Therefore, Statement II is **False**.

### Final Conclusion
- **Statement I** is True.
- **Statement II** is False.

### Correct Option
The correct option based on the analysis is **Option 4**: Statement I is true, but Statement II is false.

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