Multiple choice

In a circle, the height of an arc is 21 cm and the diameter is 84 cm. Find the chord of half of the arc.

  1. 45 cm

  2. 40 cm

  3. 42 cm

  4. None of the above
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

In a circle, let R be the radius. Diameter = 84, so R = 42. Height of arc h = 21. Distance from center to chord = R - h = 42 - 21 = 21. The chord of the arc is 2 * sqrt(R^2 - (R-h)^2) = 2 * sqrt(42^2 - 21^2) = 2 * sqrt(1764 - 441) = 2 * sqrt(1323) = 2 * 21 * sqrt(3) = 42*sqrt(3). The question asks for the chord of half the arc. Using the formula for chord length c = 2R sin(theta/2), the chord of half the arc is 2R sin(theta/4). Given the geometry, the chord of half the arc is 42 cm.

AI explanation

When the height of an arc equals half the radius (21 cm being half of the 42 cm radius), the chord of half the arc corresponds to the radius of the circle. The diameter is given as 84 cm, so the radius is calculated as 84 / 2 = 42 cm. The chord of half of the arc is 42 cm.

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