solve this---**Image Description:** * **Type:** Geometric figure. * **Main Elements:** * A circle is shown, with a shaded region. * A square is inside the circle. * Point O is labeled at the center of the circle. * One vertex of the square coincides with the point O (the center of the circle). * The vertex of the square opposite to O lies on the boundary of the circle. * The shaded region is the area of the circle minus the area of the square. **Question Stem:** In the figure shown, point O is the center of the circle. One vertex of the square lies on the circle, and the opposite vertex is point O. If the area of the shaded region is $36\pi - 18$, what is the perimeter of the square? **Options:** A) 24 B) 72 C) $12\sqrt{2}$ D) $36\sqrt{2}$

视频信息

答案文本 复制

视频字幕 复制

Let's solve this geometric problem. We have a circle with center O. A square has one vertex at O and the opposite vertex on the circle. The shaded region represents the area of the circle minus the area of the square, which equals 36 pi minus 18. We need to find the perimeter of the square. First, let's establish the relationship between the radius of the circle and the side length of the square. Let's establish the key relationship in this problem. We know that one vertex of the square is at the center O, and the opposite vertex lies on the circle. This means the diagonal of the square equals the radius of the circle. For a square with side length s, the diagonal equals s times the square root of 2. So we can write: s times square root of 2 equals r, where r is the radius. Solving for s, we get: s equals r divided by square root of 2, which can also be written as r times square root of 2 divided by 2. This relationship will help us solve for the perimeter of the square. Now, let's find the relationship between the areas. The area of the circle is pi r squared. The area of the square is s squared, which equals r squared over 2 when we substitute our expression for s. The shaded area is the area of the circle minus the area of the square, which gives us r squared times pi minus one-half. We're told that the shaded area equals 36 pi minus 18. Setting these equal, we get: 36 pi minus 18 equals r squared times pi minus one-half. Factoring the right side: 36 pi minus 18 equals 36 times pi minus one-half. Therefore, r squared equals 36, which means r equals 6. Now that we know the radius is 6, we can find the side length of the square. Using our formula, s equals r times square root of 2 divided by 2, which gives us 6 times square root of 2 divided by 2, which simplifies to 3 times square root of 2. The perimeter of the square is 4 times the side length, so the perimeter equals 4 times 3 square root of 2, which equals 12 square root of 2. Therefore, the answer is option C: 12 square root of 2. To summarize what we've learned: When a square has one vertex at the center of a circle and the opposite vertex on the circle, the diagonal of the square equals the radius of the circle. This gives us the relationship that the side length equals the radius divided by the square root of 2. The area of the shaded region is the area of the circle minus the area of the square, which equals r squared times pi minus one-half. When this area equals 36 pi minus 18, we determined that the radius is 6 and the side length is 3 square root of 2. Therefore, the perimeter of the square is 12 square root of 2, which corresponds to option C.