https://file.aiquickdraw.com/mathbot/file/1726676621536q330xsfw.jpg

The equation of a circle is given by the formula:

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

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

### Step 1: Identify the center of the circle
From the graph, we can see that the center of the circle is \((4, 2)\).

### Step 2: Find the radius
The radius is the distance from the center to any point on the circle. Using the point \((8, 5)\), we can calculate the radius as the distance between \((4, 2)\) and \((8, 5)\) using the distance formula:

\[
r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Substitute the coordinates:
\[
r = \sqrt{(8 - 4)^2 + (5 - 2)^2} = \sqrt{(4)^2 + (3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5
\]

### Step 3: Write the equation
Now that we have the center \((4, 2)\) and the radius \(r = 5\), we can write the equation of the circle:

\[
(x - 4)^2 + (y - 2)^2 = 5^2
\]

\[
(x - 4)^2 + (y - 2)^2 = 25
\]

So, the equation of the circle is:

\[
(x - 4)^2 + (y - 2)^2 = 25
\]

Would you like more details or explanations? 

Here are some related questions:
1. How do you calculate the distance between two points on a coordinate plane?
2. What is the significance of the radius in the equation of a circle?
3. Can a circle's equation be expressed in different forms?
4. How do you derive the general formula for a circle’s equation?
5. What is the role of the center and radius in determining a circle's shape?

**Tip:** The equation of a circle is always in terms of squared differences from the center, making it easy to locate the center directly from the equation.

Determine the equation of the circle graphed below. The center of the circle is at (4, 2) and a point on the circle is (8, 5).

Learn how to find the equation of a circle with a center at (4, 2) and a radius calculated from the point (8, 5). This tutorial covers the use of the distance formula and the standard form of a circle equation. Suitable for students in Grades 9-12.

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