In a circle of radius 17 cm, two parallel chords are drawn on opposite sides of a diameter. The distance between the chords is 23 cm. If the length of one chord is 16 cm, then the length of the other is:

Options

A. 23 cm
B. 30 cm
C. 15 cm
D. none of these

Answer

To solve the problem, we need to find the length of the second chord in a circle with a radius of 17 cm, given that the distance between two parallel chords is 23 cm and one of the chords has a length of 16 cm.

### Step-by-Step Solution:

1. **Understanding the Circle and Chords**:
- Let the center of the circle be O.
- Let the two parallel chords be CD and AB, where CD is 16 cm long and AB is the chord we need to find.
- The distance between the two chords is given as 23 cm.

2. **Dividing the Chord**:
- Since the chord CD is 16 cm long, we can divide it into two equal parts. Thus, each half (MD) is 8 cm (since 16 cm / 2 = 8 cm).

3. **Setting Up the Geometry**:
- Let the distance from the center O to chord CD be x cm.
- Therefore, the distance from the center O to chord AB will be (23 - x) cm, since the total distance between the chords is 23 cm.

4. **Applying the Pythagorean Theorem**:
- For chord CD:
\[ OC^2 = OM^2 + MD^2 \]
Here, OC is the radius (17 cm), OM is the distance from the center to chord CD (x cm), and MD is half the length of chord CD (8 cm).
\[ 17^2 = x^2 + 8^2 \]
\[ 289 = x^2 + 64 \]
\[ x^2 = 289 - 64 = 225 \]
\[ x = \sqrt{225} = 15 \text{ cm} \]

5. **Finding the Distance to the Other Chord**:
- Now, we can find the distance from the center O to chord AB:
\[ OB = 23 - x = 23 - 15 = 8 \text{ cm} \]

6. **Applying the Pythagorean Theorem Again**:
- For chord AB, let NB be half the length of chord AB.
\[ OB^2 = ON^2 + NB^2 \]
Here, OB is the radius (17 cm), ON is the distance from the center to chord AB (8 cm), and NB is what we need to find.
\[ 17^2 = 8^2 + NB^2 \]
\[ 289 = 64 + NB^2 \]
\[ NB^2 = 289 - 64 = 225 \]
\[ NB = \sqrt{225} = 15 \text{ cm} \]

7. **Finding the Length of Chord AB**:
- Since NB is half the length of chord AB, the full length of chord AB is:
\[ AB = 2 \times NB = 2 \times 15 = 30 \text{ cm} \]

### Final Answer:
The length of the other chord (AB) is **30 cm**.