Multiple choice

$AB$ is a chord of the circle with center $O$ and radius $r$, $OD\pm AB$ meeting $AB$ at $ D$. If $AB =8$ cm and $OD =3$ cm, then $r$ equals

  1. $7$ cm
  2. $11$ cm
  3. $5$ cm
  4. $\sqrt { 73 } $ cm
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

A perpendicular from the center to a chord bisects the chord, so AD = 8/2 = 4 cm. The radius, the 3 cm distance to the chord, and the 4 cm half-chord form a right triangle. Therefore r^2 = 3^2 + 4^2 = 25, so r = 5 cm, option C.

AI explanation

The perpendicular drawn from the center of a circle to a chord bisects the chord, meaning AD equals half of AB, which is 4 cm. This creates a right triangle OAD where the hypotenuse is the radius r, and the legs are AD of 4 cm and OD of 3 cm. By the Pythagorean theorem, r squared equals 3 squared plus 4 squared, which simplifies to 9 plus 16, giving r squared equals 25. Taking the positive square root, the radius r equals 5 cm.

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