In the given figure, the chords AB and CD are parallel and are of 36 cm and 48 cm, respectively. The distance between them is 42 cm. The diameter of the circle is _______.

Given:

Parallel chords AB = 36 cm

Parallel chords CD = 48 cm

Distance between the chords = 42 cm

Formula Used:

The perpendicular from the center of a circle to a chord bisects the chord.

Pythagoras theorem: (radius)² = (half of chord length)² + (distance from center)²

Diameter = 2 × radius

Calculation:

Let the radius of the circle be r.

Let the distance of chord AB from the center be x cm.

Then the distance of chord CD from the center will be (42 - x) cm.

For chord AB:

r² = (36/2)² + x²

⇒ r² = 18² + x²

⇒ r² = 324 + x² (Equation 1)

For chord CD:

r² = (48/2)² + (42 - x)²

⇒ r² = 24² + (42 - x)²

⇒ r² = 576 + (1764 - 84x + x²)

⇒ r² = 576 + 1764 - 84x + x²

⇒ r² = 2340 - 84x + x² (Equation 2)

Equating Equation 1 and Equation 2:

324 + x² = 2340 - 84x + x²

⇒ 324 = 2340 - 84x

⇒ 84x = 2340 - 324

⇒ 84x = 2016

⇒ x = 2016 / 84

⇒ x = 24 cm

Substitute the value of x in Equation 1:

r² = 324 + (24)²

⇒ r² = 324 + 576

⇒ r² = 900

⇒ r = \(\sqrt{900}\)

⇒ r = 30 cm

Diameter = 2 × radius = 2 × 30 = 60 cm

∴ The diameter of the circle is 60 cm.