Two circles of same radius 5 cm, intersect each other at A and B. If AB = 8 cm, then the distance between the centres is

Options

A. 10 cm
B. 4 cm
C. 6 cm
D. 8 cm

Answer

To find the distance between the centers of two intersecting circles with a radius of 5 cm and a distance of 8 cm between the intersection points A and B, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Radius of both circles (r) = 5 cm
- Distance between points A and B (AB) = 8 cm

2. **Determine the Midpoint of AB:**
- Let M be the midpoint of AB. Since AB = 8 cm, we can find AM and MB:
\[ AM = MB = \frac{AB}{2} = \frac{8 \text{ cm}}{2} = 4 \text{ cm} \]

3. **Draw the Triangles:**
- Consider the triangles formed by the centers of the circles (C1 and C2) and the points A and B. The triangles C1AB and C2AB are isosceles triangles because C1A = C1B = C2A = C2B = radius = 5 cm.

4. **Apply the Pythagorean Theorem:**
- In triangle C1AM, we can apply the Pythagorean theorem:
\[ C1A^2 = C1M^2 + AM^2 \]
- Substituting the known values:
\[ 5^2 = C1M^2 + 4^2 \]
\[ 25 = C1M^2 + 16 \]
- Rearranging gives:
\[ C1M^2 = 25 - 16 = 9 \]
- Taking the square root:
\[ C1M = \sqrt{9} = 3 \text{ cm} \]

5. **Calculate the Distance Between the Centers:**
- Since C1M and C2M are equal (both are 3 cm), the total distance between the centers C1 and C2 is:
\[ C1C2 = C1M + C2M = 3 \text{ cm} + 3 \text{ cm} = 6 \text{ cm} \]

### Final Answer:
The distance between the centers of the two circles is **6 cm**.

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