A circle has two parallel chords of lengths 6 cm and 8 cm. If the chords are 1 cm apart and the centre is on the same side of the chords, then a diameter of the circles is of length:

Options

A. 5 cm
B. 6 cm
C. 8 cm
D. 10 cm

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the Problem
We have a circle with two parallel chords of lengths 6 cm and 8 cm. The distance between the two chords is 1 cm, and both chords are on the same side of the center of the circle.

### Step 2: Define the Geometry
Let’s denote:
- The center of the circle as O.
- The chord of length 6 cm as AB.
- The chord of length 8 cm as CD.
- The distance between the two chords as 1 cm.

Since the chords are parallel, we can assume that chord AB is closer to the center O than chord CD.

### Step 3: Divide the Chords
Since AB is 6 cm long, the segments from the center O to points A and B are each 3 cm (half of 6 cm). Similarly, for chord CD which is 8 cm long, the segments from O to points C and D are each 4 cm (half of 8 cm).

### Step 4: Set Up the Coordinate System
We can set up a coordinate system:
- Let O be at (0, 0).
- Let the distance from O to chord AB be y1.
- Let the distance from O to chord CD be y2.

Since the chords are 1 cm apart, we have:
\[ y2 = y1 + 1 \]

### Step 5: Apply Pythagorean Theorem
Using the Pythagorean theorem for the triangles formed by the radius, the distance from the center to the chord, and half the length of the chord, we can write the following equations:

1. For chord AB:
\[ R^2 = y1^2 + 3^2 \quad (1) \]

2. For chord CD:
\[ R^2 = y2^2 + 4^2 \quad (2) \]

### Step 6: Substitute y2 in Terms of y1
From the distance relation, substitute \( y2 \) in equation (2):
\[ R^2 = (y1 + 1)^2 + 4^2 \]

### Step 7: Set the Equations Equal
Since both equations equal \( R^2 \), we can set them equal to each other:
\[ y1^2 + 9 = (y1 + 1)^2 + 16 \]

### Step 8: Expand and Simplify
Expanding the right side:
\[ y1^2 + 9 = y1^2 + 2y1 + 1 + 16 \]
\[ y1^2 + 9 = y1^2 + 2y1 + 17 \]

### Step 9: Solve for y1
Subtract \( y1^2 \) from both sides:
\[ 9 = 2y1 + 17 \]
\[ 2y1 = 9 - 17 \]
\[ 2y1 = -8 \]
\[ y1 = -4 \]

### Step 10: Calculate y2
Now substitute \( y1 \) back to find \( y2 \):
\[ y2 = y1 + 1 = -4 + 1 = -3 \]

### Step 11: Calculate R
Now substitute \( y1 \) back into either equation (1) or (2) to find \( R \):
Using equation (1):
\[ R^2 = (-4)^2 + 3^2 \]
\[ R^2 = 16 + 9 = 25 \]
\[ R = 5 \text{ cm} \]

### Step 12: Find the Diameter
The diameter \( D \) of the circle is given by:
\[ D = 2R = 2 \times 5 = 10 \text{ cm} \]

### Final Answer
The diameter of the circle is **10 cm**.

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