Question

Two parallel chords in a circle are at a distance of 44 cm from each other. If they lie on the opposite sides of the centre and are respectively 80 cm and 96 cm long, calculate the radius of the circle.

Answer 1

Given:

• Two parallel chords of lengths 80 cm and 96 cm in a circle.
• They lie on opposite sides of the centre.
• The distance between the chords is 44 cm.

Let:

O be the center of the circle.

OM = x be the perpendicular distance from the center O to the chord of length 80 cm.

ON = 44 – x be the perpendicular distance from O to the chord of length 96 cm.

r be the radius of the circle.

Step 1: Half-lengths of the chords:

Half of 80 cm = 40 cm.

Half of 96 cm = 48 cm.

Step 2: Use the right triangle formed by the radius, distance from center to chord, and half-chord length:

For chord of length 80 cm:

r² = x² + 40²

= x² + 1600

For chord of length 96 cm: 

r² = (44 – x)² + 48²

= (44 – x)² + 2304

Step 3: Set these equal since they both represent r²: 

x² + 1600 = (44 – x)² + 2304

x² + 1600 = 1936 – 88x + x² + 2304

x² + 1600 = x² + (1936 + 2304) – 88x

Subtract ( x² ) from both sides:

1600 = 4240 – 88x

Rearranging:

88x = 4240 – 1600 

88x = 2640

`x = 2640/88`

x = 30 cm

Step 4: Find the radius r: 

r² = x² + 40²

= 30² + 1600

= 900 + 1600

= 2500

r = `sqrt(2500)`

r = 50 cm

The radius of the circle is 50 cm.