Two circles of radii 10 cm and 8 cm intersect each other and the length of common chord is 12 cm. The distance between their centres is _______

Options

A. `sqrt7` cm
B. `3sqrt7` cm
C. `4sqrt7` cm
D. `(8+2sqrt7)` cm

Answer

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

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Radius of Circle 1 (R₁) = 10 cm
- Radius of Circle 2 (R₂) = 8 cm
- Length of the common chord (AB) = 12 cm

2. **Find Half of the Common Chord:**
- Since the common chord is bisected by the line joining the centers of the circles, we can find the half-length of the chord.
\[ AC = \frac{AB}{2} = \frac{12}{2} = 6 \text{ cm} \]

3. **Use the Right Triangle Formed:**
- In the triangle formed by the radius of Circle 1 (OA), the perpendicular from the center O to the chord (AC), and the segment AC, we can apply the Pythagorean theorem.
- For Circle 1:
\[ OA^2 = AC^2 + OC^2 \]
\[ 10^2 = 6^2 + OC^2 \]
\[ 100 = 36 + OC^2 \]
\[ OC^2 = 100 - 36 = 64 \]
\[ OC = \sqrt{64} = 8 \text{ cm} \]

4. **Repeat for Circle 2:**
- For Circle 2, we also apply the Pythagorean theorem:
\[ OB^2 = AC^2 + PC^2 \]
\[ 8^2 = 6^2 + PC^2 \]
\[ 64 = 36 + PC^2 \]
\[ PC^2 = 64 - 36 = 28 \]
\[ PC = \sqrt{28} = 2\sqrt{7} \text{ cm} \]

5. **Calculate the Distance Between the Centers:**
- The distance between the centers (OP) is the sum of OC and PC:
\[ OP = OC + PC \]
\[ OP = 8 + 2\sqrt{7} \text{ cm} \]

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

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