In a circle with centre O. AB and CD are parallel chords lying on opposite sides of a diameter parallel to them. If AB = 30 cm, CD - 48 cm and the distance between AB and CD is 27 cm, find the radius of the circle.

Answer

To find the radius of the circle given the lengths of the parallel chords AB and CD, and the distance between them, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Length of chord AB = 30 cm
- Length of chord CD = 48 cm
- Distance between chords AB and CD = 27 cm

2. **Determine Half Lengths of the Chords:**
- Since the chords are bisected by the perpendicular from the center of the circle, we can find:
- AM (half of AB) = 30 cm / 2 = 15 cm
- DN (half of CD) = 48 cm / 2 = 24 cm

3. **Set Up the Geometry:**
- Let O be the center of the circle.
- Let M be the midpoint of chord AB and N be the midpoint of chord CD.
- The distance OM + ON = 27 cm, where OM is the distance from the center to chord AB and ON is the distance from the center to chord CD.

4. **Define Variables:**
- Let OM = x cm.
- Then, ON = 27 - x cm.

5. **Apply the Pythagorean Theorem:**
- For triangle OAM:
\[ OA^2 = OM^2 + AM^2 \implies R^2 = x^2 + 15^2 \implies R^2 = x^2 + 225 \]
- For triangle ODN:
\[ OD^2 = ON^2 + DN^2 \implies R^2 = (27 - x)^2 + 24^2 \]

6. **Set the Equations Equal:**
- Since both expressions equal R², we can set them equal to each other:
\[ x^2 + 225 = (27 - x)^2 + 576 \]

7. **Expand and Simplify:**
- Expanding the right side:
\[ x^2 + 225 = 729 - 54x + x^2 + 576 \]
- Simplifying gives:
\[ 225 = 1305 - 54x \]
- Rearranging:
\[ 54x = 1305 - 225 = 1080 \implies x = \frac{1080}{54} = 20 \]

8. **Find the Radius:**
- Substitute x back into the equation for R²:
\[ R^2 = x^2 + 225 = 20^2 + 225 = 400 + 225 = 625 \]
- Therefore, R = √625 = 25 cm.

### Final Answer:
The radius of the circle is **25 cm**.