Multiple choice

In a circle of radius $10$ cm, a chord is drawn $6$ cm from the centre. If a chord half the length of the original chord were drawn, its distance in centimeters from the centre would be

  1. $\sqrt{84}$
  2. $9$
  3. $8$
  4. $3\pi $
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A Correct answer
Explanation

The radius r is 10 and the distance d1 is 6. Using Pythagoras, half the chord length is sqrt(10^2 - 6^2) = 8, so the full chord is 16. A chord half this length is 8, meaning half its length is 4. The distance d2 from the center is sqrt(10^2 - 4^2) = sqrt(100 - 16) = sqrt(84).

AI explanation

Using the perpendicular bisector property of a chord, the distance from the center squared plus half the chord length squared equals the radius squared. This gives half the original chord length as sqrt(10^2 - 6^2) = 8 cm, making the full original chord 16 cm. A chord half that length measures 8 cm, so half of this new chord is 4 cm; using the same theorem, its distance from the center is sqrt(10^2 - 4^2) = sqrt(84) cm.

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