Two parallel chords of a circle lie on the same side of the centre and are 5 cm apart. One chord has a length of 20 cm and the other chord has a length of 28 cm. Using properties of chords and perpendicular distances from the centre, what is the radius of the circle in centimetres?

Introduction / Context:
This geometry question tests knowledge of chords in a circle and how the perpendicular from the centre to a chord relates to the chord length and the radius. When two parallel chords lie on the same side of the centre, their distances from the centre and their lengths are linked through the Pythagorean relationship in right triangles formed by the radius and half chord lengths.

Given Data / Assumptions:

• There is one circle with radius r centimetres.
• The chords are parallel and on the same side of the centre.
• The distance between the chords is 5 cm.
• Longer chord length is 28 cm, so its half length is 14 cm.
• Shorter chord length is 20 cm, so its half length is 10 cm.
• Let the perpendicular distance from the centre to the longer chord be x cm, then the distance to the shorter chord is x + 5 cm.

Concept / Approach:
For any chord of a circle, radius r, and perpendicular distance d from the centre, half the chord length L/2 is given by sqrt(r^2 - d^2). This comes from a right angled triangle formed by the radius, the distance to the chord, and half the chord. We apply this formula to both chords, form two equations in r and x, and solve them simultaneously. Finally, we compute the approximate numerical value of the radius and match it with the given options.

Step-by-Step Solution:
For the longer chord: 14 = sqrt(r^2 - x^2).Square both sides: 196 = r^2 - x^2, so r^2 = x^2 + 196.For the shorter chord: 10 = sqrt(r^2 - (x + 5)^2).Square this: 100 = r^2 - (x + 5)^2, hence r^2 = 100 + (x + 5)^2.Equate the two expressions for r^2: x^2 + 196 = 100 + (x + 5)^2.Expand: x^2 + 196 = 100 + x^2 + 10x + 25, so 196 = 125 + 10x.Thus 10x = 71 and x = 7.1 cm.Now r^2 = x^2 + 196 = 7.1^2 + 196 ≈ 50.41 + 196 = 246.41, so r ≈ sqrt(246.41) ≈ 15.69 cm.

Verification / Alternative check:
Use r ≈ 15.69 and x ≈ 7.1. For the longer chord, half length is sqrt(15.69^2 - 7.1^2) which is approximately sqrt(246.41 - 50.41) = sqrt(196) = 14, so full length 28 cm. For the shorter chord, distance from centre is x + 5 = 12.1 cm. Half chord length is sqrt(15.69^2 - 12.1^2) = sqrt(246.41 - 146.41) = sqrt(100) = 10, so full length 20 cm. Both chords match the given data.

Why Other Options Are Wrong:
Values such as 14.69 cm, 16.42 cm, 17.20 cm, and 18.65 cm either produce incorrect half chord lengths or do not keep the distance between chords equal to 5 cm when substituted back into the radius relation. Only 15.69 cm satisfies both chord lengths simultaneously.

Common Pitfalls:
Common errors include mixing up which chord is closer to the centre, ignoring that the distance between chords is on the same side of the centre, or forgetting to square the entire term (x + 5). Students also sometimes approximate too early and lose accuracy. It is better to keep exact squares until the final square root step and then round to two decimal places.

Final Answer:
The radius of the circle is approximately 15.69 cm.