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The problem in the image asks to find the center-radius form of a circle, given the endpoints of its diameter \((7, 4)\) and \((-1, 6)\). Here's how to solve it step by step:

### Step 1: Find the midpoint of the diameter
The midpoint of a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is calculated as:
\[
\text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]
Substituting the values from the given endpoints \((7, 4)\) and \((-1, 6)\):
\[
\text{Midpoint} = \left( \frac{7 + (-1)}{2}, \frac{4 + 6}{2} \right) = \left( \frac{6}{2}, \frac{10}{2} \right) = (3, 5)
\]
Thus, the center of the circle is at \((3, 5)\).

### Step 2: Calculate the radius
The radius is the distance from the center to either endpoint of the diameter. The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Using the center \((3, 5)\) and one of the endpoints, say \((7, 4)\), the radius is:
\[
\text{Radius} = \sqrt{(7 - 3)^2 + (4 - 5)^2} = \sqrt{(4)^2 + (-1)^2} = \sqrt{16 + 1} = \sqrt{17}
\]
So the radius is \(\sqrt{17}\).

### Step 3: Write the equation of the circle
The equation of a circle in center-radius form is:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
where \((h, k)\) is the center and \(r\) is the radius. Substituting \((3, 5)\) for the center and \(\sqrt{17}\) for the radius, we get:
\[
(x - 3)^2 + (y - 5)^2 = 17
\]

Thus, the equation of the circle is:
\[
(x - 3)^2 + (y - 5)^2 = 17
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you find the equation of a circle if you're only given the radius and center?
2. What is the general form of the equation of a circle, and how does it differ from the center-radius form?
3. How do you calculate the distance between two points on a coordinate plane?
4. How can you find the equation of a circle if only its circumference is known?
5. What are the properties of a diameter in a circle, and how is it related to the radius?

**Tip**: For any circle-related problem, knowing the center-radius form \((x - h)^2 + (y - k)^2 = r^2\) makes it easier to graph and analyze the circle!

Find the center-radius form for the circle having the endpoints (7,4) and (-1,6) of a diameter.

Learn how to find the center-radius form of a circle given the endpoints (7, 4) and (-1, 6) of its diameter. This detailed solution includes key concepts from geometry, such as the midpoint formula, distance formula, and the equation of a circle. Ideal for students in Grades 9-10.

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