To find the radius of the circle, we use the properties of chords and the Pythagorean theorem.

Circle Radius Calculation Using Parallel Chords

Let the radius of the circle be r. Let the centre of the circle be O.

We are given two parallel chords, AB and CD, with lengths:

• AB = 6 cm
• CD = 8 cm

The distance between the chords is 7 cm, and the centre O lies between them.

Let the distance of chord AB from the centre be x cm, and the distance of chord CD from the centre be y cm.

Since the centre lies between the parallel chords, the distance between them is the sum of their distances from the centre:

$x + y = 7 \quad (1)$

Applying Chord Properties and Pythagorean Theorem

A perpendicular from the centre of a circle to a chord bisects the chord.

For chord AB:

• Half length of AB = $\frac{6}{2} = 3$ cm.
• Using the Pythagorean theorem in the right-angled triangle formed by the radius, half the chord, and the distance x:

$ \left(\frac{AB}{2}\right)^2 + x^2 = r^2 $

$ 3^2 + x^2 = r^2 $

$ 9 + x^2 = r^2 \quad (2) $

For chord CD:

• Half length of CD = $\frac{8}{2} = 4$ cm.
• Using the Pythagorean theorem:

$ \left(\frac{CD}{2}\right)^2 + y^2 = r^2 $

$ 4^2 + y^2 = r^2 $

$ 16 + y^2 = r^2 \quad (3) $

Solving for Distances and Radius

Equating equations (2) and (3) because both equal r²:

$ 9 + x^2 = 16 + y^2 $

$ x^2 - y^2 = 16 - 9 $

$ x^2 - y^2 = 7 $

Factor the left side using the difference of squares formula ($a^2 - b^2 = (a-b)(a+b)$):

$ (x - y)(x + y) = 7 $

Substitute the value of ($x + y$) from equation (1):

$ (x - y)(7) = 7 $

$ x - y = 1 \quad (4) $

Now we have a system of two linear equations with two variables (x and y):

1. $ x + y = 7 $

2. $ x - y = 1 $

Add equation (1) and (4):

$ (x + y) + (x - y) = 7 + 1 $

$ 2x = 8 $

$ x = 4 $

Substitute the value of x back into equation (1):

$ 4 + y = 7 $

$ y = 3 $

Determining the Circle's Radius

Now, substitute the value of x (or y) into equation (2) (or (3)) to find the radius r.

Using equation (2):

$ r^2 = 9 + x^2 $

$ r^2 = 9 + 4^2 $

$ r^2 = 9 + 16 $

$ r^2 = 25 $

$ r = \sqrt{25} $

$ r = 5 $

The radius of the circle is 5 cm.