Let O be the centre of the circle and AB and CD are two parallel chords on the…

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

Step-by-Step Analysis
Geometric Setup:

Let O be the center.

Let x be the perpendicular distance from O to chord AB. Since OP is perpendicular to AB, P bisects AB. Therefore, AP = AB / 2 = 10 / 2 = 5 cm.

Let y be the perpendicular distance from O to chord CD. Since OQ is perpendicular to CD, Q bisects CD. Therefore, CQ = CD / 2 = 24 / 2 = 12 cm.

Given the distance between chords PQ = 7 cm and that they are on the same side of the center, the difference in their distances from the center is |x - y| = 7. Thus, y = x + 7.

Apply the Pythagorean Theorem:
Let r be the radius of the circle. We have two right-angled triangles formed by the radius, half-chord, and perpendicular distance:

For chord AB: r² = x² + 5² => r² = x² + 25

For chord CD: r² = y² + 12² => r² = (x + 7)² + 144

Solve for x and r:
Since both expressions equal r², set them equal:
x² + 25 = (x + 7)² + 144
x² + 25 = x² + 14x + 49 + 144
25 = 14x + 193
14x = 25 - 193
14x = -168 (taking magnitude for distance, x = 12 cm)

Now, substitute x back into the radius equation:
r² = 12² + 5² = 144 + 25 = 169
r = √169 = 13 cm.

Find the Diameter:
Diameter = 2 * r = 2 * 13 = 26 cm.