solve this---4 In the following figure O is the center of circle, find the area of the shaded part. Α) 16 - 32π Β) 16π - 32 C) 32π D) 16π

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We have a circle with center O and radius 4. A square is inscribed inside the circle. We need to find the area of the shaded region, which is the area of the circle minus the area of the square. The answer choices are given as A through D. First, let's calculate the area of the circle. The formula for the area of a circle is A equals pi times r squared. With radius 4, we get A equals pi times 4 squared, which equals 16 pi. Next, we calculate the area of the inscribed square. The diagonal of the square equals the diameter of the circle, which is 2 times 4, equals 8. The area of a square using its diagonal is one half times d squared. So the area equals one half times 8 squared, which equals one half times 64, equals 32. Finally, we calculate the shaded area by subtracting the square area from the circle area. The shaded area equals 16 pi minus 32. This matches answer choice B: 16 pi minus 32. To summarize our solution: We calculated the circle area as 16 pi using the formula pi r squared. The inscribed square has a diagonal equal to the circle diameter of 8, giving us a square area of 32. The shaded area is the difference: 16 pi minus 32, which is answer choice B.