AB and CD are two chords such that AB = 10 cm , CD = 24 cm and AB // CD The distance between the chords is 17 cm . Find the radius of the circle.

Answer

To find the radius of the circle given the two parallel chords AB and CD, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Information**:
- Length of chord AB = 10 cm
- Length of chord CD = 24 cm
- Distance between the chords = 17 cm
- Chords AB and CD are parallel.

2. **Draw the Diagram**:
- Draw a circle and label the center as O.
- Draw the chord AB and chord CD such that they are parallel to each other.
- Let the perpendicular distance from the center O to chord AB be x cm, and the distance from O to chord CD will then be (17 - x) cm.

3. **Find Half-Lengths of the Chords**:
- Half of chord AB (10 cm) = 5 cm.
- Half of chord CD (24 cm) = 12 cm.

4. **Apply the Pythagorean Theorem**:
- For triangle OAB:
\[ OA^2 = OB^2 + AB^2 \]
\[ r^2 = x^2 + 5^2 \quad \text{(1)} \]
- For triangle OCD:
\[ OC^2 = OD^2 + CD^2 \]
\[ r^2 = (17 - x)^2 + 12^2 \quad \text{(2)} \]

5. **Set the Two Equations Equal**:
- Since both equations equal \( r^2 \), we can set them equal to each other:
\[ x^2 + 25 = (17 - x)^2 + 144 \]

6. **Expand and Simplify**:
- Expand the right side:
\[ x^2 + 25 = 289 - 34x + x^2 + 144 \]
- Cancel \( x^2 \) from both sides:
\[ 25 = 433 - 34x \]
- Rearranging gives:
\[ 34x = 408 \]
- Solving for x:
\[ x = \frac{408}{34} = 12 \]

7. **Substitute x back to find r**:
- Substitute \( x = 12 \) into equation (1):
\[ r^2 = 12^2 + 5^2 \]
\[ r^2 = 144 + 25 = 169 \]
- Therefore, the radius \( r \) is:
\[ r = \sqrt{169} = 13 \text{ cm} \]

### Final Answer:
The radius of the circle is **13 cm**.