Multiple choice

At one end A of a diameter AB of a circle of radius 5 cm, tangent XAY is drawn to the circle. The length of the chord CD parallel to XY and at a distance 8 cm from A is

  1. $4 cm$
  2. $5 cm$
  3. $6 cm$
  4. $8 cm$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Radius = 5. Distance from center to chord = 8 - 5 = 3. Half-length of chord = sqrt(5^2 - 3^2) = sqrt(16) = 4. Total length = 2 * 4 = 8.

AI explanation

The tangent XAY is perpendicular to the diameter AB at point A, so the distance from A to the chord CD equals the perpendicular distance from the tangent line XY to the chord CD, which is 8 cm. The center O of the circle is located 5 cm from A along the diameter, so the perpendicular distance from the center O to the chord CD is 8 minus 5, or 3 cm. Using the property that the perpendicular from the center bisects the chord, half the chord length squared equals the radius squared minus the distance squared, giving 5 squared minus 3 squared, which is 16. The square root of 16 is 4 cm for half the chord, so the full length of CD is 8 cm.

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