Multiple choice

A right angled isosceles triangle is inscribed in a circle of radius r What is the area of the remaining portion of the circle?

  1. $\displaystyle \frac{\pi r^{2}}{2}$
  2. $\displaystyle \left ( \pi -\frac{1}{2} \right )r^{2}$
  3. $\left ( \pi -1 \right )r^{2}$
  4. $\left ( \pi -2 \right )r^{2}$
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C Correct answer
Explanation

A right isosceles triangle inscribed in a circle has a hypotenuse equal to the diameter (2r). The legs are r * sqrt(2). The area of the triangle is (1/2) * base * height = (1/2) * (r * sqrt(2)) * (r * sqrt(2)) = r^2. The remaining area is the circle area minus triangle area: pi * r^2 - r^2 = (pi - 1) * r^2.

AI explanation

In a right angled isosceles triangle inscribed in a circle, the hypotenuse is the diameter of the circle, making its length 2r. Since the legs are equal, the Pythagorean theorem gives the area of the triangle as half the base times height, which is r squared. The total area of the circle is pi times r squared, so subtracting the area of the triangle leaves an area of pi times r squared minus r squared, which equals pi minus 1 times r squared.

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