The measure of the line segment joining the centre of a circle to the mid-point of a chord is :

Options

A. twice the measure of the chord
B. half the measure of the chord
C. equal to the measure of the chord
D. None of the above

Answer

To find the measure of the line segment joining the center of a circle to the midpoint of a chord, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Circle and Components**:
- Let the center of the circle be denoted as point \( O \).
- Let the chord be denoted as \( AB \).
- Let \( M \) be the midpoint of the chord \( AB \).

2. **Define the Radius and Chord Length**:
- Let the radius of the circle be \( R \).
- Let the length of the chord \( AB \) be \( L \).
- Since \( M \) is the midpoint of \( AB \), we have \( AM = MB = \frac{L}{2} \).

3. **Apply the Pythagorean Theorem**:
- In the right triangle \( OMA \), where \( OA \) is the radius \( R \), \( AM \) is \( \frac{L}{2} \), and \( OM \) is the segment we want to find.
- According to the Pythagorean theorem, we can write:
\[ OA^2 = OM^2 + AM^2 \]
- Substituting the known values:
\[ R^2 = OM^2 + \left(\frac{L}{2}\right)^2 \]

4. **Rearranging the Equation**:
- Rearranging the equation to solve for \( OM^2 \):
\[ OM^2 = R^2 - \left(\frac{L}{2}\right)^2 \]

5. **Simplifying the Expression**:
- Expanding \( \left(\frac{L}{2}\right)^2 \):
\[ OM^2 = R^2 - \frac{L^2}{4} \]
- To find \( OM \), take the square root:
\[ OM = \sqrt{R^2 - \frac{L^2}{4}} \]

### Final Result:
The measure of the line segment joining the center of the circle to the midpoint of the chord is:
\[ OM = \sqrt{R^2 - \frac{L^2}{4}} \]