Multiple choice

Angle ABC is $40^0$ and the area of the circle is 81$\pi$. If CB is a diameter of the circle, how long is arc AXC?

  1. $4\pi$ units
  2. $2\pi$ units
  3. $\pi$ units
  4. $\dfrac {\pi}{2}$ units
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A Correct answer
Explanation

Area = 81pi = pi * r^2, so r = 9. Diameter CB = 18. In triangle ABC, angle B = 180 - 90 - 40 = 50 degrees. The arc length AXC corresponds to the central angle subtended by chord AC. Since CB is diameter, angle CAB = 90. Arc AC = (angle ABC / 180) * pi * r is incorrect; the angle at center is 2 * angle ABC = 80 degrees. Arc length = (80/360) * 2 * pi * 9 = 4pi.

AI explanation

From the area of the circle pi r^2 = 81 pi, the radius r is found to be 9 units. Since CB is a diameter, the angle subtended by the arc AXC at the center is twice the inscribed angle ABC, making it 2 * 40 = 80 degrees. The length of arc AXC is calculated using the arc length formula (80/360) * 2 * pi * 9, which simplifies to (2/9) * 18 pi. Therefore, the length of arc AXC is 4 pi units.

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