Final answer:
The center of the circle with a radius of 3 and points (-5, 2) and (-5, 8) lies on it is at (-5, 5). The two points are on a vertical line, indicating a diameter, and the midpoint of this diameter is the center.
Step-by-step explanation:
To find the center of a circle when given two points on the circle and the radius, you can use the concept that the radius is perpendicular to the tangent of the circle at any point. Here, we are given two points, (-5, 2) and (-5, 8), and we are told the radius of the circle is 3.
Since the x-coordinates of the points are the same, it suggests that they are vertically aligned, and hence the line that connects these two points is a diameter of the circle. The center of the circle lies halfway between these two points on this diameter. To find the midpoint (center of the circle), you average the y-coordinates: (2 + 8) / 2 = 10 / 2 = 5. The x-coordinate of the center remains the same since we are on a vertical line. Thus, the center of the circle is (-5, 5).
Now we must verify if this center and the given radius match the provided points. The distance from the center to the given points should be equal to the radius. The distance from the point (-5, 5) to the point (-5, 8) is 3 units (vertical distance), which is the same as the given radius. Therefore, the center of the circle is correct.