Two chords AB and CD of lengths 24 cm and 10 cm respectively of a circle are parallel. If the chords lie on the same side of the centre and distance between them is 7 cm, find the length of a diameter of the circle.

Answer

To find the diameter of the circle given the lengths of two parallel chords and the distance between them, we can follow these steps:

### Step 1: Understand the Problem
We have two chords AB and CD of lengths 24 cm and 10 cm, respectively. The distance between the two chords is 7 cm. We need to find the diameter of the circle.

### Step 2: Bisect the Chords
Since the chords are parallel, we can drop perpendiculars from the center of the circle to each chord. Let’s denote the midpoint of AB as M and the midpoint of CD as N.

- Length of AB = 24 cm, so AM = MB = 12 cm (since M bisects AB).
- Length of CD = 10 cm, so CN = ND = 5 cm (since N bisects CD).

### Step 3: Set Up the Geometry
Let the distance from the center of the circle to chord AB be \( h \) and the distance from the center to chord CD be \( h + 7 \) (since the distance between the two chords is 7 cm).

### Step 4: Apply the Pythagorean Theorem
We can apply the Pythagorean theorem in triangles formed by the radius, the half-length of the chord, and the distance from the center to the chord.

For chord AB:
\[ r^2 = h^2 + 12^2 \]
\[ r^2 = h^2 + 144 \quad \text{(1)} \]

For chord CD:
\[ r^2 = (h + 7)^2 + 5^2 \]
\[ r^2 = (h + 7)^2 + 25 \quad \text{(2)} \]

### Step 5: Expand Equation (2)
Expanding equation (2):
\[ r^2 = (h^2 + 14h + 49) + 25 \]
\[ r^2 = h^2 + 14h + 74 \quad \text{(3)} \]

### Step 6: Set Equations Equal
Since both equations (1) and (3) equal \( r^2 \), we can set them equal to each other:
\[ h^2 + 144 = h^2 + 14h + 74 \]

### Step 7: Simplify the Equation
Cancel \( h^2 \) from both sides:
\[ 144 = 14h + 74 \]
Subtract 74 from both sides:
\[ 70 = 14h \]
Divide by 14:
\[ h = 5 \]

### Step 8: Find the Radius
Now substitute \( h = 5 \) back into equation (1) to find the radius:
\[ r^2 = 5^2 + 144 \]
\[ r^2 = 25 + 144 \]
\[ r^2 = 169 \]
Taking the square root:
\[ r = 13 \, \text{cm} \]

### Step 9: Calculate the Diameter
The diameter \( D \) of the circle is given by:
\[ D = 2r \]
\[ D = 2 \times 13 = 26 \, \text{cm} \]

### Final Answer
The length of the diameter of the circle is **26 cm**.