A.) To find the center of the circle, we can use the midpoint formula. The midpoint formula states that the coordinates of the midpoint of a line segment with endpoints (x1, y1) and (x2, y2) are given by:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
In this case, the endpoint coordinates are (-5, -11) and (11, 19), and we want to find the midpoint. Plugging the values into the midpoint formula, we get:
Midpoint = ((-5 + 11)/2, (-11 + 19)/2)
= (6/2, 8/2)
= (3, 4)
Therefore, the ordered pair (3, 4) represents the center of the circle.
B.) To find the length of the radius of the circle, we can use the distance formula. The distance formula states that the distance between two points (x1, y1) and (x2, y2) is given by:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
In this case, the two points are (-5, -11) and (3, 4). Plugging the values into the distance formula, we get:
Distance = sqrt((3 - (-5))^2 + (4 - (-11))^2)
= sqrt((8)^2 + (15)^2)
= sqrt(64 + 225)
= sqrt(289)
= 17
Therefore, the length of the radius of the circle is 17 units.
C.) The particle can never pass through a point with an x-coordinate of 24 because the given information states that the particle starts at (-5, -11), moves halfway around the circle, and ends up at (11, 19). The x-coordinate of the particle changes from -5 to 11, which is a total change of 16 units. This means the x-coordinate can only go up to 11 + 16 = 27 or down to -5 - 16 = -21. It cannot reach the x-coordinate of 24, as it is outside the range of possible values for the particle's position on the circular path. Therefore, the particle can never pass through a point with an x-coordinate of 24 as long as it stays on the circular path.