The radius of the circle is 10 cm and the perpendicular distance from the chord to the centre is 6 cm. The length of the chord is:

Correct — C. The perpendicular from the centre to a chord bisects it, so the radius, that perpendicular and half the chord make a right-angled triangle.

Radius (the hypotenuse) = 10 cm
Perpendicular distance = 6 cm
Half-chord² = 10² − 6² = 100 − 36 = 64
Half-chord = 8 cm

Chord = 2 × 8 = 16 cm → option (c).

Why the others are wrong

• (a)
  12 cm — 12 cm is the answer to a different distance. Half of 12 is 6, but 6 is the perpendicular you were given, not the half-chord; a 12 cm chord would sit 8 cm from this centre.
• (b)
  18 cm — Too long for a chord 6 cm out. An 18 cm chord has half-length 9 and would lie only √(100 − 81) ≈ 4.36 cm from the centre — the further a chord is from the centre, the shorter it gets.
• (d)
  14 cm — 7 is not the half-chord. A 14 cm chord splits into halves of 7, which would place it √(100 − 49) ≈ 7.14 cm from the centre rather than the 6 cm the stem gives.

One theorem carries this whole family of questions: a perpendicular dropped from the centre to a chord bisects the chord. That is what licenses the half.

It turns radius, distance and half-chord into the three sides of a right triangle, tied by r² = d² + (c⁄2)². Any two of them give the third, which is why the same figure can be asked from any corner.

Note the direction of the relationship: as d grows, c shrinks. A chord through the centre, with d = 0, is the diameter and the longest chord there is.

Key facts

• A perpendicular from the centre of a circle to a chord bisects that chord.
• For radius r, centre-to-chord distance d and chord length c, r² = d² + (c⁄2)².
• 6, 8, 10 is a Pythagorean triple, which is why these particular numbers were chosen.
• The longest chord of a circle is the diameter, here 20 cm, so any answer above 20 is impossible on sight.

Common traps

• Treating the 6 cm as half the chord instead of the distance from the centre
• Finding 8 and forgetting to double it
• Putting the diameter, 20, in the hypotenuse slot where the radius, 10, belongs

No directly related past PYQ was found.