The lengths of two parallel chords of a circle are 6 cm and 8 cm. If the smaller chord is at distance 4 cm from the centre, what is the distance of the other chord from the centre?

Options

A. `5 cm`
B. `3 cm`
C. `2 cm`
D. `4 cm`

Explanation

To solve the problem step by step, we will follow a structured approach:

### Step 1: Understand the problem
We have two parallel chords in a circle. The lengths of the chords are given as 6 cm and 8 cm. The smaller chord (6 cm) is at a distance of 4 cm from the center of the circle. We need to find the distance of the larger chord (8 cm) from the center.

### Step 2: Draw the circle and chords
Let's draw a circle with center O. Label the smaller chord as AB (6 cm) and the larger chord as CD (8 cm).

### Step 3: Find the half-length of the smaller chord
Since the chord AB is 6 cm long, the half-length of AB is:
\[ AP = PB = \frac{6}{2} = 3 \text{ cm} \]
where P is the midpoint of chord AB.

### Step 4: Use the right triangle formed by the radius and the chord
In triangle OPB (where O is the center, P is the midpoint of chord AB, and B is one endpoint of the chord), we can apply the Pythagorean theorem:
\[ OB^2 = OP^2 + PB^2 \]
Here, \( OP = 4 \text{ cm} \) (the distance from the center to the chord), \( PB = 3 \text{ cm} \), and \( OB \) is the radius \( R \).

### Step 5: Substitute the values and solve for R
Substituting the known values:
\[ R^2 = 4^2 + 3^2 \]
\[ R^2 = 16 + 9 = 25 \]
\[ R = \sqrt{25} = 5 \text{ cm} \]

### Step 6: Find the distance of the larger chord from the center
Let Q be the midpoint of the larger chord CD. We need to find the distance OQ from the center to the chord CD. The half-length of chord CD is:
\[ CQ = \frac{8}{2} = 4 \text{ cm} \]

Using the Pythagorean theorem in triangle OQC:
\[ OC^2 = OQ^2 + CQ^2 \]
Substituting \( OC = R = 5 \text{ cm} \) and \( CQ = 4 \text{ cm} \):
\[ 5^2 = OQ^2 + 4^2 \]
\[ 25 = OQ^2 + 16 \]
\[ OQ^2 = 25 - 16 = 9 \]
\[ OQ = \sqrt{9} = 3 \text{ cm} \]

### Final Answer
The distance of the other chord (CD) from the center is **3 cm**.

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