Geometry problem

In the following figure, an isosceles triangle ABC is perfectly enclosed in a semi-circle of radius r. If AB is a diameter of the semi-circle, which of the following expressions shows the combined area of the shaded regions in terms of r?

A. r²(π – 1) / 2

B. r²(π – 2)

C. r²(π – 2) / 2

D. r²(π – 1) 

The combined area of the shaded regions is determined from subtracting the area of the triangle from the area of the semi-circle. AB = 2r, and this segment must pass through the center of the circle. This means that the height of the triangle (to the vertex C) must be equal to the radius r. Thus the area of the triangle is 2r(r)/2 = r². Area of the semi-circle is πr²/2. Thus the combined area of the shaded regions is πr²/2 – r² = r²(π – 2) / 2. The correct answer choice is (C).

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