Final answer:
Only Option B, stating that the midpoint of (4, 5) and (12, 11) is the center of the circle, is correct. The center is at (8, 8), and the other options can be ruled out after performing the required calculations, such as the distance formula for Option C.
Step-by-step explanation:
The endpoints of a diameter of a circle are given as (4, 5) and (12, 11). To address the question regarding these points, we will consider the provided options one by one.
• Option A suggests that (4, 5) and (12, 11) are endpoints of the radius, not the diameter. However, this is incorrect as the endpoints of a diameter are indeed located at the circumference of the circle, diametrically opposite each other.
• Option B states that the midpoint of (4, 5) and (12, 11) is the center of the circle. This is true because the midpoint of the diameter of a circle is always the center of the circle. The midpoint can be calculated using the average of the x-coordinates and the average of the y-coordinates of the endpoints [(4+12)/2, (5+11)/2], which results in (8, 8).
• Option C claims that the circle has a radius of 5 units. To verify this, we need to calculate the distance between the endpoints using the distance formula d = √[(x_{2}-x_{1})^2 + (y_{2}-y_{1})^2]. However, substituting the given points into the distance formula, we get a different value for the radius.
• Option D gives the equation of the circle as (x - 8)^2 + (y - 8)^2 = 50. This equation is derived by using the midpoint of the diameter as the center and the square of the length of radius as the radius squared (r^2) in the standard equation of a circle ((x-h)^2 + (y-k)^2 = r^2).
After calculating the necessary values, we can determine which options are correct. The center of the circle and the correct expression for the circle's equation are most pertinent to the question.