In a circle of radius 5 cm, AB and CD are two parallel chords of lengths 8 cm and 6 cm respectively. Calculate the distance between the chords if they are
(i) on the same side of the centre.
(ii) on the opposite sides of the centre.

Answer

To solve the problem, we need to calculate the distance between two parallel chords AB and CD in a circle of radius 5 cm. The lengths of the chords are 8 cm and 6 cm respectively. We will find the distance between the chords in two scenarios: (i) when they are on the same side of the center, and (ii) when they are on opposite sides of the center.

### Step-by-Step Solution:

#### Part (i): Chords on the Same Side of the Center

1. **Draw the Circle and Chords**:
- Draw a circle with center O and radius 5 cm.
- Draw chord AB of length 8 cm and chord CD of length 6 cm, both on the same side of the center.

2. **Find the Midpoints of the Chords**:
- Since the chords are parallel, draw perpendiculars from the center O to each chord.
- Let M be the midpoint of AB and N be the midpoint of CD.

3. **Calculate the Lengths of the Segments**:
- For chord AB (length 8 cm), the half-length is 4 cm (AM = MB = 4 cm).
- For chord CD (length 6 cm), the half-length is 3 cm (CN = ND = 3 cm).

4. **Apply the Pythagorean Theorem**:
- In triangle OMA (where OA is the radius):
\[ OA^2 = OM^2 + AM^2 \implies 5^2 = OM^2 + 4^2 \]
\[ 25 = OM^2 + 16 \implies OM^2 = 9 \implies OM = 3 \text{ cm} \]
- In triangle ONC:
\[ OC^2 = ON^2 + CN^2 \implies 5^2 = ON^2 + 3^2 \]
\[ 25 = ON^2 + 9 \implies ON^2 = 16 \implies ON = 4 \text{ cm} \]

5. **Calculate the Distance Between the Chords**:
- The distance between the chords is the difference between OM and ON:
\[ \text{Distance} = ON - OM = 4 \text{ cm} - 3 \text{ cm} = 1 \text{ cm} \]

#### Part (ii): Chords on Opposite Sides of the Center

1. **Draw the Circle and Chords**:
- Again, draw a circle with center O and radius 5 cm.
- Draw chord AB of length 8 cm on one side and chord CD of length 6 cm on the opposite side of the center.

2. **Find the Midpoints of the Chords**:
- As before, draw perpendiculars from the center O to each chord.
- Let M be the midpoint of AB and N be the midpoint of CD.

3. **Calculate the Lengths of the Segments**:
- For chord AB, AM = 4 cm.
- For chord CD, CN = 3 cm.

4. **Apply the Pythagorean Theorem**:
- In triangle OMA:
\[ OA^2 = OM^2 + AM^2 \implies 5^2 = OM^2 + 4^2 \]
\[ 25 = OM^2 + 16 \implies OM^2 = 9 \implies OM = 3 \text{ cm} \]
- In triangle ONC:
\[ OC^2 = ON^2 + CN^2 \implies 5^2 = ON^2 + 3^2 \]
\[ 25 = ON^2 + 9 \implies ON^2 = 16 \implies ON = 4 \text{ cm} \]

5. **Calculate the Distance Between the Chords**:
- The distance between the chords is the sum of OM and ON:
\[ \text{Distance} = OM + ON = 3 \text{ cm} + 4 \text{ cm} = 7 \text{ cm} \]

### Final Answers:
- (i) The distance between the chords on the same side of the center is **1 cm**.
- (ii) The distance between the chords on opposite sides of the center is **7 cm**.