Two non-intersecting circles one lying inside another arc of diameters a and b. The minimum distance between their circumferences is c. The distance between their centres is

Options

A. `a - b - c`
B. `a + b - c`
C. `(1)/(2) (a - b - c)`
D. `(1)/(2) (a - b) - c`

Answer

To find the distance between the centers of two non-intersecting circles, one lying inside the other with diameters A and B, and a minimum distance of C between their circumferences, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Radii of the Circles:**
- The radius of the larger circle (with diameter A) is given by:
\[ R = \frac{A}{2} \]
- The radius of the smaller circle (with diameter B) is given by:
\[ r = \frac{B}{2} \]

2. **Understanding the Minimum Distance Between Circumferences:**
- The minimum distance between the circumferences of the two circles is given as C. This means that if we take the distance between the centers of the two circles and subtract the radii of both circles, we should get C:
\[ \text{Distance between centers} - R - r = C \]

3. **Express the Distance Between the Centers:**
- Let \( D \) be the distance between the centers of the two circles. We can express this as:
\[ D - R - r = C \]
- Rearranging this equation gives us:
\[ D = C + R + r \]

4. **Substituting the Values of R and r:**
- Substitute the values of R and r into the equation:
\[ D = C + \frac{A}{2} + \frac{B}{2} \]

5. **Combining the Terms:**
- Combine the terms to simplify:
\[ D = C + \frac{A + B}{2} \]

### Final Answer:
Thus, the distance between the centers of the two circles is:
\[ D = C + \frac{A + B}{2} \]