A chord of length 14cm is at a distance of 6 cm from the center of the circle. The length of another chord at a distance of 2cm from the center of the circle is

Options

A. 12 cm
B. 14 cm
C. 16 cm
D. 18 cm

Answer

To solve the problem, we need to find the length of a chord that is at a distance of 2 cm from the center of the circle, given that another chord of length 14 cm is at a distance of 6 cm from the center. We can use the properties of circles and right triangles to find the solution.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Length of chord CD = 14 cm
- Distance from center O to chord CD (OM) = 6 cm
- Distance from center O to chord AB (ON) = 2 cm

2. **Find the Midpoint of Chord CD:**
- Let M be the midpoint of chord CD. Since M is the midpoint, we can find the length of segment MC.
- Since CD = 14 cm, then MC = CD/2 = 14/2 = 7 cm.

3. **Apply the Pythagorean Theorem in Triangle OMC:**
- In triangle OMC, we have:
\[ OM^2 + MC^2 = OC^2 \]
- Substitute the known values:
\[ 6^2 + 7^2 = OC^2 \]
- Calculate:
\[ 36 + 49 = OC^2 \]
\[ OC^2 = 85 \]
- Therefore, the radius \( R \) of the circle is:
\[ R = \sqrt{85} \]

4. **Find the Length of Chord AB:**
- Let N be the midpoint of chord AB. We need to find the length of segment BN.
- In triangle ONB, we apply the Pythagorean theorem:
\[ ON^2 + BN^2 = OB^2 \]
- Substitute the known values:
\[ 2^2 + BN^2 = R^2 \]
- Since \( R^2 = 85 \):
\[ 4 + BN^2 = 85 \]
- Rearranging gives:
\[ BN^2 = 85 - 4 = 81 \]
- Therefore, \( BN = \sqrt{81} = 9 \) cm.

5. **Calculate the Length of Chord AB:**
- Since N is the midpoint of AB, we have:
\[ AB = 2 \times BN = 2 \times 9 = 18 \text{ cm} \]

### Final Answer:
The length of chord AB is **18 cm**.