Multiple choice

Find the area of the circle if the area of an isosceles right triangle inscribed in it is 18 $\displaystyle cm^{2}$

  1. 16$\displaystyle \pi $
  2. 21$\displaystyle \pi $
  3. 18$\displaystyle \pi $
  4. 24$\displaystyle \pi $
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C Correct answer
Explanation

For an isosceles right triangle inscribed in a circle, the hypotenuse is the diameter. Area = 1/2 * base * height. If legs are 'a', area = a^2/2 = 18, so a^2 = 36, a = 6. Hypotenuse = sqrt(6^2 + 6^2) = sqrt(72) = 6*sqrt(2). Radius = 3*sqrt(2). Area of circle = pi * r^2 = pi * (3*sqrt(2))^2 = 18pi.

AI explanation

For an isosceles right triangle inscribed in a circle, the hypotenuse is the diameter of the circle. If the equal legs are x, the area is (1/2)x^2 = 18, giving x^2 = 36 so x = 6 cm. The hypotenuse is x*sqrt(2) = 6*sqrt(2) cm, making the radius half of that at 3*sqrt(2) cm. Using the area of a circle formula, pi*r^2 = pi(3*sqrt(2))^2 = 18*pi.

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