Endpoints of a diameter are (6,−12)
 and (10,4)


To find the center and radius of a circle given the endpoints of its diameter, follow these steps:

1. **Find the center of the circle**:
   The center of the circle is the midpoint of the diameter. The midpoint between two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated as:

   \[
   \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
   \]

   For the points \((6, -12)\) and \((10, 4)\):

   \[
   \text{Midpoint} = \left( \frac{6 + 10}{2}, \frac{-12 + 4}{2} \right) = \left( \frac{16}{2}, \frac{-8}{2} \right) = (8, -4)
   \]

   So, the center of the circle is \((8, -4)\).

2. **Find the radius of the circle**:
   The radius is half the length of the diameter. First, we calculate the length of the diameter using the distance formula between two points:

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

   Substituting the points \((6, -12)\) and \((10, 4)\):

   \[
   \text{Distance} = \sqrt{(10 - 6)^2 + (4 + 12)^2} = \sqrt{(4)^2 + (16)^2} = \sqrt{16 + 256} = \sqrt{272} \approx 16.49
   \]

   The diameter is approximately 16.49 units, so the radius is:

   \[
   \text{Radius} = \frac{16.49}{2} \approx 8.245 \, \text{units}
   \]

### Summary:
- **Center of the circle**: \((8, -4)\)
- **Radius of the circle**: \(\approx 8.245\)

Would you like more details on any part of the process?

Here are 5 related questions:
1. How do you find the equation of the circle using the center and radius?
2. Can the distance formula be derived geometrically?
3. How does the midpoint formula apply in 3D coordinates?
4. How would the problem change if one endpoint were unknown?
5. How can you find the slope of the diameter from the two endpoints?

**Tip**: Always double-check signs when working with coordinates to avoid errors in midpoint or distance calculations!

Endpoints of a diameter are (6,−12) and (10,4)

Discover how to find the center and radius of a circle when given the endpoints of its diameter. Using key concepts in geometry and coordinate systems, this detailed guide covers the midpoint and distance formulas with real data points like (6, -12) and (10, 4). Perfect for students in Grades 9-10.

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