answer this---Question Number: 34 Chart/Diagram Description: * Type: Geometric figure (Circle with a sector). * Main Elements: * A circle is shown. * Point C is labeled at the center of the circle. * A sector of the circle is shaded. * The central angle of the shaded sector is indicated and labeled as 100°. * Two radii forming the central angle are shown. One radius extends horizontally to the right from the center. Question Stem: Point C is the center of the circle above. What fraction of the area of the circle is the area of the shaded region? Options: [No options are provided in the image]

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In this problem, we need to find what fraction of the circle's area is represented by the shaded sector. The key insight is that the area of a sector is proportional to its central angle. Since the central angle of our sector is 100 degrees, and a full circle is 360 degrees, the fraction of the area is simply the ratio of these angles. So we calculate 100 degrees divided by 360 degrees, which equals 10 over 36. Simplifying this fraction by dividing both numerator and denominator by 2, we get 5 over 18. Therefore, the shaded region represents 5/18 of the circle's area. Let's solve this problem step by step. We know that the area of a sector is related to the area of the full circle by the ratio of the central angle to 360 degrees. This gives us the formula: Area of Sector equals theta divided by 360 degrees times the Area of Circle. In our problem, the central angle theta is 100 degrees. To find what fraction of the circle's area is represented by the sector, we divide the sector area by the circle area. This simplifies to theta divided by 360 degrees. Substituting our angle, we get 100 degrees divided by 360 degrees, which equals 100 over 360. Simplifying this fraction, we get 10 over 36, which further simplifies to 5 over 18. Therefore, the shaded region represents 5/18 of the circle's area. Now, let's focus on simplifying our fraction. We started with 100 degrees divided by 360 degrees, which gives us 100 over 360. To simplify this, we can divide both the numerator and denominator by 10, resulting in 10 over 36. We can simplify further by finding the greatest common divisor of 10 and 36, which is 2. Dividing both the numerator and denominator by 2, we get 5 over 18 as our final answer. We can verify this is correct by multiplying: 5 times 18 equals 90, and 10 times 36 equals 360. The fractions are equivalent. Looking at our visual representation, we can see that 5 out of 18 equal sectors of the circle are highlighted, representing the fraction 5/18 of the circle's area. This confirms our answer. Let's summarize what we've learned about finding the fraction of a circle's area represented by a sector. First, the area of a sector is directly proportional to its central angle. To find what fraction of the circle's area is represented by a sector, we divide the sector angle by the full circle angle of 360 degrees. In our problem, we had a sector with a central angle of 100 degrees. So the fraction is 100 degrees divided by 360 degrees. We then simplified this fraction by finding common factors. The fraction 100/360 simplifies to 10/36, which further simplifies to 5/18. Therefore, the shaded region represents 5/18 of the circle's area. This approach works for any sector: just divide the central angle by 360 degrees and simplify the resulting fraction. Let's review the key takeaways from this problem. First, the area of a sector is directly proportional to its central angle. To find what fraction of a circle's area is represented by a sector, we divide the sector angle by the full circle angle of 360 degrees. In our specific problem with a 100-degree sector, the fraction is 100 degrees divided by 360 degrees, which simplifies to 5/18 of the circle. When simplifying fractions, it's important to find the greatest common divisor of the numerator and denominator. This method works for any sector: simply divide the central angle by 360 degrees and simplify the resulting fraction. Remember that this relationship between angles and areas is a fundamental property of circles and is used in many geometric calculations. The answer to our original question is that the shaded region represents 5/18 of the circle's area.