segment AB represents the diameter of a circle. if point A is located at (8,6) and Point B is located at (-2,6). write the equation of the circle
To find the center of the circle, we can find the midpoint of the segment AB. The midpoint formula is given by:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
Using the coordinates of A and B, the midpoint is:
Midpoint = ((8 + -2)/2, (6 + 6)/2)
= (6/2, 12/2)
= (3, 6)
So, the center of the circle is (3, 6).
The radius of the circle is half the length of AB. The formula to find the length of a line segment AB is:
Length AB = √((x2 - x1)^2 + (y2 - y1)^2)
Using the coordinates of A and B, the length of AB is:
Length AB = √((-2 - 8)^2 + (6 - 6)^2)
= √((-10)^2 + 0^2)
= √(100 + 0)
= √100
= 10
Therefore, the radius of the circle is 10/2 = 5.
The equation of a circle with center (h, k) and radius r is given by:
(x - h)^2 + (y - k)^2 = r^2
Substituting the values of (h, k) = (3, 6) and r = 5, we get:
(x - 3)^2 + (y - 6)^2 = 5^2
(x - 3)^2 + (y - 6)^2 = 25
So, the equation of the circle is (x - 3)^2 + (y - 6)^2 = 25.