Name the diameter, radius, center, and the equation of the circle with a diameter that has endpoints at (-3, 4) and (5, -2).

the center is at the midpoint of diameter: (1,1)

radius = 1/2 √(8^2+6^2) = 5

(x-1)^2 + (y-1)^2 = 25

To find the diameter, radius, center, and the equation of the circle, we first need to determine the distance between the two endpoints of the diameter.

Step 1: Find the distance between the two endpoints using the distance formula.
The distance formula is given by:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Using the endpoints (-3, 4) and (5, -2), we can substitute the values into the formula:
d = sqrt((5 - (-3))^2 + (-2 - 4)^2)
= sqrt((5 + 3)^2 + (-2 - 4)^2)
= sqrt(8^2 + (-6)^2)
= sqrt(64 + 36)
= sqrt(100)
= 10

So, the diameter of the circle is 10 units.

Step 2: Find the radius.
The radius of a circle is half the diameter. Therefore, the radius is:
radius = diameter / 2
= 10 / 2
= 5

So, the radius of the circle is 5 units.

Step 3: Find the center of the circle.
The center of the circle can be found by finding the midpoint of the diameter. The midpoint formula is given by:
(h, k) = ((x1 + x2) / 2, (y1 + y2) / 2)

Using the endpoints (-3, 4) and (5, -2), we can substitute the values into the formula:
(h, k) = ((-3 + 5) / 2, (4 - 2) / 2)
= (2 / 2, 2 / 2)
= (1, 1)

So, the center of the circle is at the point (1, 1).

Step 4: Write the equation of the circle.
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 into the equation:
(x - 1)^2 + (y - 1)^2 = 5^2
(x - 1)^2 + (y - 1)^2 = 25

Therefore, the equation of the circle is (x - 1)^2 + (y - 1)^2 = 25.