A diameter of a circle has endpoints A(-4,2) and B(3,2). Find the center of the circle, radius, and write an equation for the circle.
the center is at the midpoint of AB: (-1/2,2)
the radius is 1/2 the diameter: 7/2
so now you can easily write the equation:
(x + 1/2)^2 + (y-2)^2 = (7/2)^2
Thanks so much!!! oobleck
To find the center of the circle, we need to find the midpoint of the diameter. The midpoint of a line segment with endpoints (x1, y1) and (x2, y2) can be found using the midpoint formula:
Midpoint (M) = ((x1 + x2) / 2, (y1 + y2) / 2)
In this case, the endpoints of the diameter are A(-4, 2) and B(3, 2), so we can substitute these coordinates into the midpoint formula:
Midpoint (M) = ((-4 + 3) / 2, (2 + 2) / 2)
= (-1/2, 2)
Therefore, the center of the circle is C(-1/2, 2).
To find the radius of the circle, we need to find the distance between the center of the circle and one of the endpoints of the diameter. The distance formula is used to find the distance between two points (x1, y1) and (x2, y2):
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Using the coordinates of the center C(-1/2, 2) and one endpoint A(-4, 2), we can substitute them into the distance formula:
Distance = sqrt((-4 - (-1/2))^2 + (2 - 2)^2)
= sqrt((-4 + 1/2)^2 + 0^2)
= sqrt((-7/2)^2)
= sqrt(49/4)
= 7/2
So, the radius of the circle is 7/2.
Finally, to write the equation for the circle, we use the formula:
(x - h)^2 + (y - k)^2 = r^2
Where (h, k) represents the center of the circle and r represents the radius.
In this case, the center of the circle is C(-1/2, 2) and the radius is 7/2, so substituting these values in the equation, we get:
(x - (-1/2))^2 + (y - 2)^2 = (7/2)^2
(x + 1/2)^2 + (y - 2)^2 = 49/4
Therefore, the equation of the circle is (x + 1/2)^2 + (y - 2)^2 = 49/4.