The endpoints of a diameter of a circle are (3, 4) and (7, –2). What is an equation of the circle?

So the midpoint is (5, -1)

and the equation would be

(x-5)^2 + *y+1)^2 = r^2

plug in one of the end points of the diameter to get r^2

check your answer by plugging in the other point, you should get the same r^2 value

To find the equation of the circle with the given endpoints of a diameter, we can use the midpoint formula. Let's first find the coordinates of the midpoint of the diameter.

Midpoint formula:
The midpoint of a line segment with endpoints (x₁, y₁) and (x₂, y₂) is given by the coordinates:

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

Using the given endpoints (3, 4) and (7, -2), we can find the coordinates of the midpoint:

x = (3 + 7) / 2 = 10/2 = 5
y = (4 + (-2)) / 2 = 2/2 = 1

Therefore, the coordinates of the midpoint are (5, 1).

Now, let's find the radius of the circle by calculating the distance between one of the endpoints and the midpoint. We can use the distance formula:

Distance formula:
The distance between two points (x₁, y₁) and (x₂, y₂) is given by:

Distance = sqrt((x₂ - x₁)² + (y₂ - y₁)²)

Using the coordinates of one endpoint (3, 4) and the midpoint (5, 1), we can find the radius:

r = sqrt((5 - 3)² + (1 - 4)²)
r = sqrt(2² + (-3)²)
r = sqrt(4 + 9)
r = sqrt(13)

The radius of the circle is sqrt(13).

Finally, we can write the equation of the circle using the standard form, which is:

(x - h)² + (y - k)² = r²

where (h, k) is the center of the circle, and r is the radius.

Plugging in the values, we have:

(x - 5)² + (y - 1)² = (sqrt(13))²
(x - 5)² + (y - 1)² = 13

So, the equation of the circle is (x - 5)² + (y - 1)² = 13.