Certainly! Let's work through the steps to find the equation of the circle given the endpoints of its diameter, which are (18, 10) and (-2, 10).
1. Find the Midpoint of the Diameter (Center of the Circle):
The midpoint of a line segment with endpoints [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] can be found using the midpoint formula:
[tex]\[
\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)
\][/tex]
Plugging in our points (18, 10) and (-2, 10):
[tex]\[
\left(\frac{18 + (-2)}{2}, \frac{10 + 10}{2}\right) = \left(\frac{16}{2}, \frac{20}{2}\right) = (8, 10)
\][/tex]
So, the center of the circle is [tex]\((8, 10)\)[/tex].
2. Find the Radius of the Circle:
The radius is half the length of the diameter. To find the length of the diameter, we can use the distance formula for the points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex]:
[tex]\[
\text{distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Using our points (18, 10) and (-2, 10):
[tex]\[
\text{diameter} = \sqrt{((-2) - 18)^2 + (10 - 10)^2} = \sqrt{(-20)^2 + 0^2} = \sqrt{400} = 20
\][/tex]
The radius [tex]\(r\)[/tex] is half of the diameter:
[tex]\[
r = \frac{20}{2} = 10
\][/tex]
3. Write the Equation of the Circle:
The standard form of the equation of a circle with center [tex]\((h, k)\)[/tex] and radius [tex]\(r\)[/tex] is:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
Substituting [tex]\(h = 8\)[/tex], [tex]\(k = 10\)[/tex], and [tex]\(r = 10\)[/tex], we get:
[tex]\[
(x - 8)^2 + (y - 10)^2 = 10^2
\][/tex]
Simplifying the right-hand side:
[tex]\[
(x - 8)^2 + (y - 10)^2 = 100
\][/tex]
Therefore, the equation of the circle is:
[tex]\[
(x - 8)^2 + (y - 10)^2 = 100
\][/tex]
