Multiple choice

The radius of two concentric circles is 5 cm and 13 cm, respectively. The tangent to the smaller circle is the chord to the greater circle. What is the length (in cm) of that chord?

  1. 12

  2. 20

  3. 22

  4. 24
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D Correct answer
Explanation

A right triangle is formed by the radius of the smaller circle (5 cm), half the chord, and the radius of the larger circle (13 cm). Using Pythagoras, half-chord = sqrt(13^2 - 5^2) = sqrt(169 - 25) = sqrt(144) = 12. The full chord length is 12 * 2 = 24 cm.

AI explanation

The perpendicular drawn from the common center to the tangent of the smaller circle bisects the chord of the larger circle and equals the smaller radius of 5 cm. Using the Pythagorean theorem in the right triangle formed by the larger radius of 13 cm, the smaller radius of 5 cm, and half the chord, we find half the chord length is sqrt(13^2 - 5^2) = sqrt(169 - 25) = sqrt(144) = 12 cm. The full length of the chord is 2 * 12 = 24 cm.

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