Two parallel chords of a circle , 12 cm and 16 cm long are on the same of the centre. The distance between them is 2 cm. Find the radius of the circle.

Explanation

To solve the problem of finding the radius of a circle with two parallel chords of lengths 12 cm and 16 cm, which are on the same side of the center and have a distance of 2 cm between them, we can follow these steps:

### Step-by-Step Solution:

1. **Draw the Circle and Chords**:
- Draw a circle and label the center as O.
- Draw the longer chord (16 cm) and label its endpoints as A and B.
- Draw the shorter chord (12 cm) and label its endpoints as C and D.
- Place the chords such that they are parallel and on the same side of the center, with a distance of 2 cm between them.

2. **Identify the Midpoints**:
- The midpoint of chord AB (16 cm) is M, and the midpoint of chord CD (12 cm) is N.
- Since AB is 16 cm long, AM = MB = 8 cm.
- Since CD is 12 cm long, CN = ND = 6 cm.

3. **Establish the Distance Between Chords**:
- Let the distance from the center O to chord AB be y cm.
- Therefore, the distance from O to chord CD will be (y + 2) cm because the distance between the two chords is 2 cm.

4. **Apply the Pythagorean Theorem**:
- For triangle OMA (right triangle):
\[ OA^2 = OM^2 + AM^2 \]
\[ R^2 = y^2 + 8^2 \]
\[ R^2 = y^2 + 64 \quad \text{(Equation 1)} \]

- For triangle ONC (right triangle):
\[ OC^2 = ON^2 + CN^2 \]
\[ R^2 = (y + 2)^2 + 6^2 \]
\[ R^2 = (y + 2)^2 + 36 \]
\[ R^2 = y^2 + 4y + 4 + 36 \]
\[ R^2 = y^2 + 4y + 40 \quad \text{(Equation 2)} \]

5. **Set the Equations Equal**:
- Since both expressions equal \( R^2 \), we can set them equal to each other:
\[ y^2 + 64 = y^2 + 4y + 40 \]

6. **Simplify the Equation**:
- Cancel \( y^2 \) from both sides:
\[ 64 = 4y + 40 \]
- Rearranging gives:
\[ 64 - 40 = 4y \]
\[ 24 = 4y \]
\[ y = 6 \]

7. **Find the Radius**:
- Substitute \( y \) back into Equation 1 to find \( R \):
\[ R^2 = y^2 + 64 \]
\[ R^2 = 6^2 + 64 \]
\[ R^2 = 36 + 64 \]
\[ R^2 = 100 \]
\[ R = \sqrt{100} = 10 \text{ cm} \]

### Final Answer:
The radius of the circle is **10 cm**.