A circle has a diameter with endpoints at (1, 6) and (7, 6). Which of the following points is on the circle?
Correct: A
First, find the center of the circle, which is the midpoint of the diameter.
Center (h, k) = ((1 + 7)/2, (6 + 6)/2) = (8/2, 12/2) = (4, 6).
Next, find the radius. The radius is the distance from the center to one of the endpoints. Let's use (1, 6).
Radius r = sqrt((1 - 4)^2 + (6 - 6)^2)
r = sqrt((-3)^2 + (0)^2)
r = sqrt(9) = 3.
The equation of the circle is (x - 4)^2 + (y - 6)^2 = 3^2, or (x - 4)^2 + (y - 6)^2 = 9.
Now, check each choice:
A) For (4, 9): (4 - 4)^2 + (9 - 6)^2 = 0^2 + 3^2 = 0 + 9 = 9. This point is on the circle.
B) For (1, 3): (1 - 4)^2 + (3 - 6)^2 = (-3)^2 + (-3)^2 = 9 + 9 = 18 ≠ 9.
C) For (7, 3): (7 - 4)^2 + (3 - 6)^2 = (3)^2 + (-3)^2 = 9 + 9 = 18 ≠ 9.
D) For (4, 6): This is the center of the circle, not a point on the circle (unless the radius is 0, which it isn't). (4-4)^2 + (6-6)^2 = 0+0 = 0 ≠ 9.
Therefore, (4, 9) is on the circle.