PQ and RS are two parallel chords of a circle whose centre is O and radius is 10 cm. If PQ = 16 cm and RS = 12 cm, find the distance between PQ and RS, if they lie :
(i) On the same side of the centre O.
(ii) On opposite side of the centre O.

Options

A. 2, 14 cm
B. 4, 14 cm
C. 2, 12 cm
D. 4, 12 cm

Answer

To solve the problem, we need to find the distance between two parallel chords PQ and RS in a circle with center O and radius 10 cm. We will solve this for both cases: when the chords are on the same side of the center and when they are on opposite sides.

### Given:
- Radius of the circle (r) = 10 cm
- Length of chord PQ = 16 cm
- Length of chord RS = 12 cm

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

**For chord PQ:**
1. Since PQ is 16 cm long, the half-length of PQ is:
\[ \frac{PQ}{2} = \frac{16}{2} = 8 \text{ cm} \]
2. Using the Pythagorean theorem in triangle OAQ (where A is the midpoint of PQ):
\[ OA^2 + AQ^2 = OQ^2 \]
Here, \(OQ = 10\) cm (radius), and \(AQ = 8\) cm (half-length of PQ).
\[ OA^2 + 8^2 = 10^2 \]
\[ OA^2 + 64 = 100 \]
\[ OA^2 = 100 - 64 = 36 \]
\[ OA = \sqrt{36} = 6 \text{ cm} \]

**For chord RS:**
1. Since RS is 12 cm long, the half-length of RS is:
\[ \frac{RS}{2} = \frac{12}{2} = 6 \text{ cm} \]
2. Using the Pythagorean theorem in triangle OBS (where B is the midpoint of RS):
\[ OB^2 + BS^2 = OS^2 \]
Here, \(OS = 10\) cm (radius), and \(BS = 6\) cm (half-length of RS).
\[ OB^2 + 6^2 = 10^2 \]
\[ OB^2 + 36 = 100 \]
\[ OB^2 = 100 - 36 = 64 \]
\[ OB = \sqrt{64} = 8 \text{ cm} \]

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

**(i) When PQ and RS are on the same side of O:**
- The distance between the two chords is:
\[ \text{Distance} = OB - OA = 8 \text{ cm} - 6 \text{ cm} = 2 \text{ cm} \]

**(ii) When PQ and RS are on opposite sides of O:**
- The distance between the two chords is:
\[ \text{Distance} = OA + OB = 6 \text{ cm} + 8 \text{ cm} = 14 \text{ cm} \]

### Final Answers:
- (i) The distance between PQ and RS on the same side of O is **2 cm**.
- (ii) The distance between PQ and RS on opposite sides of O is **14 cm**.