AB and CD are two parallel chords drawn in a circle with centre O. The distance between the two chords is 21 cm. If the lengths of AB and CD are 24 cm and 18 cm, respectively, then the radius of the circle is equal to:

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Given two parallel chords AB and CD in a circle with center O. The distance between the chords is 21 cm. Lengths of chords are AB = 24 cm and CD = 18 cm.

Step 1: Let the radius of the circle be R.

Step 2: Since AB and CD are chords, draw perpendiculars from center O to these chords. Let the perpendicular distances from O to AB and CD be d1 and d2 respectively.

Step 3: The distance between the two chords is 21 cm, so |d1 - d2| = 21.

Step 4: For chord AB of length 24 cm, half-length = 12 cm. Using right triangle formed by radius, half chord, and perpendicular distance:
R^2 = d1^2 + 12^2 = d1^2 + 144

Step 5: For chord CD of length 18 cm, half-length = 9 cm.
R^2 = d2^2 + 9^2 = d2^2 + 81

Step 6: Since both equal R^2,
d1^2 + 144 = d2^2 + 81
=> d1^2 - d2^2 = 81 - 144 = -63

Step 7: Use the identity a^2 - b^2 = (a - b)(a + b)
(d1 - d2)(d1 + d2) = -63

Step 8: Given |d1 - d2| = 21, so (d1 - d2) = ±21

If (d1 - d2) = 21, then (d1 + d2) = -63 / 21 = -3 (not possible since distances are positive)

If (d1 - d2) = -21, then (d1 + d2) = -63 / -21 = 3

Step 9: Solve system:
d1 - d2 = -21
d1 + d2 = 3

Adding,
2d1 = -18 => d1 = -9 (not possible as distance can't be negative)

Try swapping signs:
Let d2 - d1 = 21
Then (d1 + d2) = -63 / 21 = -3 (not possible)

Try d2 - d1 = -21
Then (d1 + d2) = -63 / -21 = 3

System:
d2 - d1 = -21
d1 + d2 = 3

Add:
2d2 = -18 => d2 = -9 (not possible)

Step 10: Since distances can't be negative, consider that the chords lie on opposite sides of the center O. So the distance between chords is d1 + d2 = 21.

Step 11: Now, from step 6:
d1^2 - d2^2 = -63

Rewrite as:
(d1 - d2)(d1 + d2) = -63

Given d1 + d2 = 21

So,
(d1 - d2) * 21 = -63
=> d1 - d2 = -3

Step 12: Solve system:
d1 + d2 = 21
d1 - d2 = -3

Add:
2d1 = 18 => d1 = 9

Then d2 = 21 - 9 = 12

Step 13: Calculate radius R:
R^2 = d1^2 + 12^2 = 9^2 + 144 = 81 + 144 = 225
=> R = sqrt(225) = 15

Check with d2:
R^2 = d2^2 + 9^2 = 12^2 + 81 = 144 + 81 = 225
=> R = 15

Step 14: The radius is 15 cm.

Step 15: However, 15 cm is an option but let's verify the distance between chords is 21 cm.

Since chords are parallel and on opposite sides of center, distance between chords = d1 + d2 = 9 + 12 = 21 cm (matches given).

Therefore, radius = 15 cm.

Final answer: 15 cm.

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