Multiple choice

Area if a circle in which a chord of length $\sqrt { 2 }$ makes an analog $\dfrac { \pi }{ 2 } $ at the center is

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

For a chord of length c subtending angle θ at the center, c = 2r sin(θ/2). Here, sqrt(2) = 2r sin(π/4) = r sqrt(2), so r = 1 and the area is π.

AI explanation

Let the radius of the circle be r. The chord of length sqrt(2) subtends an angle of pi/2 at the center, so using the chord length formula 2r sin(theta/2), we have 2r sin(pi/4) = sqrt(2). This simplifies to 2r(1/sqrt(2)) = sqrt(2), meaning r = 1. The area of the circle is pi * r^2 = pi * 1^2 = pi.

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