The length of two parallel chords of a circle are 6 cm and 8 cm . The radius of the circle is 5 cm. Find the distance btween them if :
(i) chords are on the same side of the centre.
(ii) chords are on the opposite side of the centre.

Explanation

To solve the problem, we need to find the distance between two parallel chords of a circle given their lengths and the radius of the circle. We will solve the problem for both cases: when the chords are on the same side of the center and when they are on opposite sides.

### Given:
- Length of chord AB = 6 cm
- Length of chord CD = 8 cm
- Radius of the circle (r) = 5 cm

### Step 1: Find the distances from the center to each chord.

1. **For chord AB:**
- The length of chord AB is 6 cm. Therefore, half of this length is:
\[ AM = MB = \frac{6}{2} = 3 \text{ cm} \]
- Let ON be the perpendicular distance from the center O to chord AB. Using the Pythagorean theorem in triangle OMB:
\[ OB^2 = OM^2 + MB^2 \]
Substituting the known values:
\[ 5^2 = OM^2 + 3^2 \]
\[ 25 = OM^2 + 9 \]
\[ OM^2 = 25 - 9 = 16 \]
\[ OM = 4 \text{ cm} \]

2. **For chord CD:**
- The length of chord CD is 8 cm. Therefore, half of this length is:
\[ CN = ND = \frac{8}{2} = 4 \text{ cm} \]
- Let ON be the perpendicular distance from the center O to chord CD. Using the Pythagorean theorem in triangle ONC:
\[ OC^2 = ON^2 + NC^2 \]
Substituting the known values:
\[ 5^2 = ON^2 + 4^2 \]
\[ 25 = ON^2 + 16 \]
\[ ON^2 = 25 - 16 = 9 \]
\[ ON = 3 \text{ cm} \]

### Step 2: Calculate the distance between the chords.

1. **If the chords are on the same side of the center:**
- The distance NM between the chords is given by:
\[ NM = OM - ON = 4 - 3 = 1 \text{ cm} \]

2. **If the chords are on opposite sides of the center:**
- The distance NM between the chords is given by:
\[ NM = OM + ON = 4 + 3 = 7 \text{ cm} \]

### Final Answers:
- (i) Distance between the chords when they are on the same side = **1 cm**
- (ii) Distance between the chords when they are on opposite sides = **7 cm**