The points ​(StartRoot 7 EndRoot7​,negative 1−1​) and ​(StartRoot 3 EndRoot3​,33​) are endpoints of the diameter of a circle. Determine the center of the circle.

To determine the center of the circle when given the endpoints of the diameter, we use the midpoint formula. The midpoint of the diameter is the center of the circle.

Given points:
- \((\sqrt{7}, -1)\)
- \((\sqrt{3}, 3)\)

The midpoint formula is:

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

Substituting the given points into the formula:

\[
\text{Center} = \left(\frac{\sqrt{7} + \sqrt{3}}{2}, \frac{-1 + 3}{2}\right)
\]

Let's calculate each coordinate:

1. **x-coordinate**:
   \[
   \frac{\sqrt{7} + \sqrt{3}}{2}
   \]

2. **y-coordinate**:
   \[
   \frac{-1 + 3}{2} = \frac{2}{2} = 1
   \]

So, the center of the circle is:

\[
\left(\frac{\sqrt{7} + \sqrt{3}}{2}, 1\right)
\]

Would you like more details on this process or have any questions?

Here are 5 related questions for further exploration:

1. How do you find the radius of the circle given the same endpoints?
2. What is the equation of the circle given these points?
3. How can you find the length of the diameter of the circle?
4. What are the general properties of a circle’s diameter?
5. How does the midpoint formula relate to finding the center of a circle?

**Tip:** Remember that the midpoint formula is a valuable tool for finding the center of various geometric shapes, not just circles!

Discover how to find the center of a circle using the midpoint formula with detailed steps. Given endpoints (sqrt(7), -1) and (sqrt(3), 3), learn the mathematical process to determine the circle's center. Suitable for high school students.

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