A chord of a circle has a length of 2√21 cm. The distance of the chord from the center is 5 cm. What is the radius of the circle?

Correct — B.
The perpendicular from the centre bisects the chord.
Half-chord: 2√21 ÷ 2 = √21 cm

Pythagoras: r² = (√21)² + 5² = 21 + 25 = 46
Take the root: r = √46 cm
Estimate: 6.78² ≈ 45.97, just under 46, so r ≈ 6.78 cm → option (b)

Why the others are wrong

• (a)
  6 cm — 6² = 36, not 46. A 6 cm radius with the chord 5 cm from the centre leaves a half-chord of √(36 − 25) = √11, so the chord would be 2√11 cm, not 2√21 cm.
• (c)
  8.89 cm — 8.89² ≈ 79, far above r² = 46. A radius that large would put a 2√21 cm chord about 7.6 cm from the centre, not 5 cm.
• (d)
  9 cm — 9² = 81, so a 2√21 cm chord would sit √(81 − 21) = √60 ≈ 7.7 cm from the centre, not 5 cm.

The perpendicular from the centre to a chord bisects it. That gives a right triangle: half the chord and the centre-to-chord distance are the legs, and the radius is the hypotenuse.

So r² = (half-chord)² + (distance)². Given any two of radius, chord and distance, this finds the third.

Here the half-chord is the surd √21, so squaring it gives a whole number and the arithmetic stays clean until the final root.

Key facts

• The perpendicular from the centre to a chord bisects the chord
• r² = (half-chord)² + (distance from centre)²
• √46 ≈ 6.78, since 6.78² ≈ 45.97

Common traps

• Using the full chord 2√21 as a leg instead of half of it, which gives r = √109 ≈ 10.44
• Squaring 2√21 as 42 instead of 4 × 21 = 84, forgetting that the coefficient is squared too

No directly related past PYQ was found.