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COMEDK202510 May 2025Evening ShiftMathematicsProperties of TrianglesActual

The least area of a circle circumscribing any right-angle triangle of area 9 sq units is

Options

  1. A4.5 sq units
  2. B9 sq units
  3. Csq units
  4. D9 sq units

Correct answer

D. 9 sq units

Step-by-step solution

Let the sides of the right-angled triangle be a , b , and c , where c is the hypotenuse. The area of the triangle is given by A = 1 2 ab = 9 . The radius R of the circumcircle of a right-angled triangle is half the length of the hypotenuse, so R = c 2 . By the Pythagorean theorem, c^2 = a^2 + b^2 . Since a^2 + b^2 2ab , we have c^2 2ab . Substituting ab = 2A = 18 , we get c^2 2 18 = 36 . The radius R satisfies R^2 = c^2 4 36 4 = 9 . The area of the circumcircle is S = R^2 . Since R^2 9 , the minimum area is S = 9 =

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