solve this---2 A C Ο The circle above with center O has a circumference of 36. What is the length of minor arc AC ? A) 9 B) 12 C) 18 D) 36

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In this problem, we have a circle with center O and a circumference of 36 units. We need to find the length of the minor arc AC. Notice that the angle AOC is a right angle, which is 90 degrees. To solve this problem, we need to understand that the length of an arc is proportional to the central angle that subtends it. To calculate the length of arc AC, we need to follow three steps. First, we identify the central angle AOC, which is 90 degrees as shown by the right angle. Second, we find what fraction of the full circle this represents. Since a full circle is 360 degrees, the fraction is 90 divided by 360, which equals one-fourth. Finally, we calculate the arc length by multiplying this fraction by the circumference. So, the arc length equals one-fourth times 36, which gives us 9 units. Therefore, the answer is A) 9. Let's understand the general formula for calculating arc length. The arc length equals the central angle in degrees divided by 360 degrees, multiplied by the circumference of the circle. Alternatively, we can express this as theta divided by 360 degrees times 2 pi r, where r is the radius. In our example, we have a central angle of 90 degrees and a circumference of 36. Applying the formula, we get 90 divided by 360 times 36, which equals one-fourth times 36, giving us 9 units. As we can see, if we change the central angle, the arc length changes proportionally. Therefore, the correct answer is A) 9. Let's summarize what we've learned. We were given a circle with a circumference of 36 units. We identified that the central angle AOC is 90 degrees, which is a right angle. To find the arc length, we used the formula: arc length equals the central angle divided by 360 degrees, multiplied by the circumference. Substituting our values, we get 90 degrees divided by 360 degrees, times 36, which equals one-fourth times 36, giving us 9 units. Therefore, the length of the minor arc AC is 9 units, and the correct answer is A) 9. Let's review the key takeaways from this problem. First, we learned that arc length is directly proportional to the central angle. The formula relates the arc length to the fraction of the full circle and its circumference. Specifically, for a circle with circumference C and central angle θ, the arc length is calculated as θ divided by 360 degrees, multiplied by C. This relationship works universally for any arc on any circle, regardless of size. Understanding this concept is fundamental for solving problems involving circular motion, sectors, and other geometric applications. As we saw in our problem, when the central angle was 90 degrees, or one-fourth of a full circle, the arc length was one-fourth of the circumference, which gave us 9 units.