Multiple choice

Find the area of circle in sq. cms, in which the chord of length 18 cm is at the distance of half of the radius of circle from the centre of circle:

  1. $108 \pi$
  2. $54 \pi$
  3. $27 \pi$
  4. $81 \pi$
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A Correct answer
Explanation

Let radius be R. Distance from center to chord is R/2. Using Pythagorean theorem on the triangle formed by radius, half-chord, and distance: R^2 = (R/2)^2 + 9^2. R^2 = R^2/4 + 81. 3/4 * R^2 = 81. R^2 = 108. Area = pi * R^2 = 108 * pi.

AI explanation

The perpendicular distance from the center to the chord bisects the 18 cm chord, creating a right triangle with a leg of 9 cm. Since this distance is half the radius, we can use the Pythagorean theorem where 9 squared plus the quantity 0.5 times r squared equals r squared. Solving 81 equals 0.75 times r squared gives the radius squared as 108. Therefore, the area of the circle is pi times r squared, which is 108 pi.

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