To determine which statement is true about Cynthia’s position at the center of a circular playground, let’s analyze the nature of circles and the implications of standing at the center.
Given two statements:
A. The distance from pole 1 to pole 3 is equal to the distance from Cynthia to pole 1.
B. The distance from pole 1 to pole 3 is equal to the distance from pole 3 to pole 2.
Here’s a step-by-step solution:
1. Identify Cynthia's position: Cynthia is at the center of the circle. This means that she is equidistant from all points on the perimeter of the circle.
2. Nature of distances in a circle:
The distance from any point on the circumference (like a pole) to the center of the circle is called the radius. So, the distance from Cynthia to any pole on the circumference is a radius (r).
The statement that the distance from Cynthia to pole 1 equals the radius (r) is immediately true for any pole, including pole 1, pole 2, and pole 3.
3. Examine Statement A:
Statement A compares the distance from pole 1 to pole 3 with the distance from Cynthia to pole 1.
Since distances from Cynthia to any pole are radii of the circle (r), the distance from Cynthia to pole 1 is indeed r.
The distance from one point on the circle to another (such as from pole 1 to pole 3) is usually not a radius because it depends on their angular separation around the circle, which can vary.
4. Assessing the validity of Statement A:
The distance from Cynthia to pole 1 is certainly a radius (r), not necessarily equal to the distance between pole 1 to pole 3, which could be anything from close to zero (if they are near each other) to close to the circumference itself if they are nearly diametrically opposite each other.
5. Examine Statement B:
Statement B compares the distance from pole 1 to pole 3 with the distance from pole 3 to pole 2.
For distances on the perimeter, their equality or inequality depends entirely on their specific positions (angles) on the circle's circumference.
* Without additional information about the relative positions of the poles, you cannot determine a consistent relationship between these distances.
Given the analysis, Statement A is indeed true because the constant distance from Cynthia to any pole is simply the radius of the circle, while Statement B cannot be verified without additional information about the poles' positions on the playground.
Thus, the correct statement and final answer is:
1 (which corresponds to Statement A being true).
