In a circle of radius 17 cm, two parallel chords of lengths 30 cm and 16 cm are drawn. If both the chords are on the same side of the centre, then the distance between the chords is a

Options

A. 9 cm
B. 7 cm
C. 23 cm
D. 11 cm

Answer

To solve the problem of finding the distance between two parallel chords in a circle, we can follow these steps:

### Step 1: Understand the Problem
We have a circle with a radius of 17 cm and two parallel chords of lengths 30 cm and 16 cm. We need to find the distance between these two chords, which are on the same side of the center of the circle.

### Step 2: Identify the Chords and Their Midpoints
Let the chords be named AB (30 cm) and CD (16 cm). The midpoints of these chords will be denoted as M (for AB) and N (for CD).

### Step 3: Calculate Half-Lengths of the Chords
- The length of chord AB is 30 cm, so half of this length is:
\[ AM = \frac{30}{2} = 15 \text{ cm} \]
- The length of chord CD is 16 cm, so half of this length is:
\[ CN = \frac{16}{2} = 8 \text{ cm} \]

### Step 4: Apply the Pythagorean Theorem
Since the radius of the circle is the hypotenuse of the right triangle formed by the radius, the distance from the center to the chord, and half the length of the chord, we can apply the Pythagorean theorem.

1. For chord AB:
\[ OA^2 = OM^2 + AM^2 \]
Substituting the known values:
\[ 17^2 = OM^2 + 15^2 \]
\[ 289 = OM^2 + 225 \]
\[ OM^2 = 289 - 225 = 64 \]
\[ OM = \sqrt{64} = 8 \text{ cm} \]

2. For chord CD:
\[ OC^2 = ON^2 + CN^2 \]
Substituting the known values:
\[ 17^2 = ON^2 + 8^2 \]
\[ 289 = ON^2 + 64 \]
\[ ON^2 = 289 - 64 = 225 \]
\[ ON = \sqrt{225} = 15 \text{ cm} \]

### Step 5: Calculate the Distance Between the Chords
The distance between the two chords (NM) can be found by subtracting the distance from the center to chord AB (OM) from the distance from the center to chord CD (ON):
\[ NM = ON - OM = 15 \text{ cm} - 8 \text{ cm} = 7 \text{ cm} \]

### Final Answer
The distance between the two chords is **7 cm**.

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