The diameter of a circle is 20 cm. There are two parallel chords of length 16 cm . And 12 cm. Find the distance between these chords if chords are on the:
(i) same side
(ii) opposite side of the centre.

Explanation

To solve the problem, we need to find the distance between two parallel chords of lengths 16 cm and 12 cm in a circle with a diameter of 20 cm. We will consider two cases: when the chords are on the same side of the center and when they are on opposite sides.

### Given:
- Diameter of the circle (d) = 20 cm
- Radius of the circle (r) = d/2 = 10 cm
- Length of the first chord (AB) = 16 cm
- Length of the second chord (CD) = 12 cm

### Step 1: Find the distance from the center to each chord when they are on the same side.

#### For chord AB (length 16 cm):
1. The chord is bisected by a perpendicular from the center.
2. Half the length of chord AB = 16 cm / 2 = 8 cm.
3. Let the distance from the center O to chord AB be \( h_1 \).
4. Using the Pythagorean theorem in triangle OAQ (where Q is the midpoint of AB):
\[ OA^2 = OQ^2 + AQ^2 \]
\[ 10^2 = h_1^2 + 8^2 \]
\[ 100 = h_1^2 + 64 \]
\[ h_1^2 = 100 - 64 = 36 \]
\[ h_1 = \sqrt{36} = 6 \text{ cm} \]

#### For chord CD (length 12 cm):
1. Half the length of chord CD = 12 cm / 2 = 6 cm.
2. Let the distance from the center O to chord CD be \( h_2 \).
3. Using the Pythagorean theorem in triangle OCP (where P is the midpoint of CD):
\[ OC^2 = OP^2 + CP^2 \]
\[ 10^2 = h_2^2 + 6^2 \]
\[ 100 = h_2^2 + 36 \]
\[ h_2^2 = 100 - 36 = 64 \]
\[ h_2 = \sqrt{64} = 8 \text{ cm} \]

### Step 2: Find the distance between the two chords on the same side.
- The distance between the two chords \( PQ \) when they are on the same side is:
\[ PQ = h_2 - h_1 = 8 \text{ cm} - 6 \text{ cm} = 2 \text{ cm} \]

### Step 3: Find the distance from the center to each chord when they are on opposite sides.

#### For chord AB (length 16 cm):
- The distance \( h_1 \) remains the same as calculated before:
\[ h_1 = 6 \text{ cm} \]

#### For chord CD (length 12 cm):
- The distance \( h_2 \) also remains the same as calculated before:
\[ h_2 = 8 \text{ cm} \]

### Step 4: Find the distance between the two chords on opposite sides.
- The distance between the two chords \( PQ \) when they are on opposite sides is:
\[ PQ = h_1 + h_2 = 6 \text{ cm} + 8 \text{ cm} = 14 \text{ cm} \]

### Final Answers:
- (i) Distance between the chords on the same side: **2 cm**
- (ii) Distance between the chords on opposite sides: **14 cm**