A segment with endpoints C(-3, 3)and D(7, 3) is the diameter of a circle.

a. What is the center of the circle?
b. What is the length of the radius?
c. What is the circumference of the circle?
d. What is the equation of the circle?

look at the points, a diameter is along the y axis. Its length is y1-y2=10

(a) center O is at midpoint of CD (O = (C+D)/2)

(b) radius r is the distance OC or OD (use your distance formula)
(c) circumference = 2pi*r
(d) if O is (h,k) then the equation is (x-h)^2 + (y-k)^2 = r^2 (from a and b above)

a. To find the center of the circle, we can find the midpoint of the segment CD. The midpoint formula is given by the average of the x-coordinates and the average of the y-coordinates of the endpoints.

The x-coordinate of the midpoint is (x₁ + x₂) / 2 = (-3 + 7) / 2 = 4 / 2 = 2.
The y-coordinate of the midpoint is (y₁ + y₂) / 2 = (3 + 3) / 2 = 6 / 2 = 3.

Therefore, the center of the circle is (2, 3).

b. The radius of the circle is half the length of the diameter. Since the diameter is the distance between points C(-3, 3) and D(7, 3) horizontally, the distance between the x-coordinates of the endpoints of the segment is the length of the diameter.

The length of the diameter is |x₁ - x₂| = |-3 - 7| = |-10| = 10.
The radius is half the length of the diameter, so the length of the radius is 10 / 2 = 5.

Therefore, the length of the radius is 5.

c. The circumference of the circle can be found using the formula C = 2πr, where r is the radius of the circle.

Substituting the value of the radius (5) into the formula, we have:
C = 2π(5) = 10π.

Therefore, the circumference of the circle is 10π.

d. The equation of a circle with center (h, k) and radius r is given by the equation:
(x - h)² + (y - k)² = r².

Substituting the values of the center (2, 3) and the radius (5) into the equation, we have:
(x - 2)² + (y - 3)² = 5².

Therefore, the equation of the circle is (x - 2)² + (y - 3)² = 25.

To find the answers to the questions, we need to understand the properties of circles and use the given information.

a. The center of a circle can be found by taking the midpoint of the line segment formed by its diameter. In this case, the coordinates of the endpoints of the diameter are C(-3, 3) and D(7, 3). The midpoint formula is given by:

Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)

So, the x-coordinate of the center is:

x-coordinate = (-3 + 7) / 2 = 4 / 2 = 2

The y-coordinate remains the same at 3. Therefore, the center of the circle is at (2, 3).

b. The length of the radius can be found by determining the distance between one of the endpoints of the diameter and the center of the circle. We can use the distance formula, which is given by:

Distance = √((x2 - x1)^2 + (y2 - y1)^2)

Using the coordinates of the center (2, 3) and one endpoint (C(-3, 3)), we have:

Distance = √((2 - (-3))^2 + (3 - 3)^2) = √((5)^2 + (0)^2) = √(25) = 5

So, the length of the radius is 5.

c. The circumference of a circle can be calculated using the formula:

Circumference = 2πr

Where r is the radius of the circle. In this case, the radius is 5, so:

Circumference = 2π(5) = 10π

Therefore, the circumference of the circle is 10π.

d. The equation of a circle can be found using the coordinates of its center (h, k) and the radius (r). The general equation of a circle is:

(x - h)^2 + (y - k)^2 = r^2

Using the coordinates of the center (2, 3) and the radius 5, we have:

(x - 2)^2 + (y - 3)^2 = 5^2

Simplifying further:

(x - 2)^2 + (y - 3)^2 = 25

Therefore, the equation of the circle is (x - 2)^2 + (y - 3)^2 = 25.