Multiple choice

$O$ is the centre of the circle. $AB$ and $CD$ are two chords of the circle. $OM \perp AB$ and $ON \perp CD$. If $OM = ON = 3$ cm and $AM = BM = 4.5$ cm, then $CD$ is equal to

  1. $8$ cm
  2. $9$ cm
  3. $10$ cm

  4. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Chords equidistant from the center are equal in length. Since OM = ON, AB = CD. Given AM = BM = 4.5, AB = 9. Therefore, CD = 9.

AI explanation

Since OM is equal to ON and both are perpendicular distances from the center to the respective chords, the two chords AB and CD must be equidistant from the center and therefore congruent. Using the Pythagorean theorem on triangle OMA, the radius OA is the square root of (3 squared plus 4.5 squared), which equals the square root of 29.25. Applying this radius to the right triangle formed with chord CD, half of CD is the square root of (29.25 minus 3 squared), which is the square root of 20.25, or 4.5 cm. The full length of CD is twice 4.5 cm, which equals 9 cm.

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