To determine certain properties of a circle given the endpoints of its diameter, [tex]\((3, 6)\)[/tex] and [tex]\((-1, 4)\)[/tex], we will follow these steps:
### 1. Find the Midpoint of the Diameter
The midpoint of the points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] can be calculated using the midpoint formula:
[tex]\[ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \][/tex]
Plugging in the given points:
[tex]\[ \text{Midpoint} = \left( \frac{3 + (-1)}{2}, \frac{6 + 4}{2} \right) \][/tex]
[tex]\[ = \left( \frac{2}{2}, \frac{10}{2} \right) \][/tex]
[tex]\[ = (1.0, 5.0) \][/tex]
So, the center of the circle (midpoint of the diameter) is [tex]\((1.0, 5.0)\)[/tex].
### 2. Calculate the Length of the Diameter
To find the length of the diameter, we use the distance formula between 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 the given points:
[tex]\[ \text{Distance} = \sqrt{((-1) - 3)^2 + (4 - 6)^2} \][/tex]
[tex]\[ = \sqrt{(-4)^2 + (-2)^2} \][/tex]
[tex]\[ = \sqrt{16 + 4} \][/tex]
[tex]\[ = \sqrt{20} \][/tex]
[tex]\[ = 4.47213595499958 \][/tex]
The length of the diameter is approximately [tex]\(4.472\)[/tex].
### 3. Determine the Radius
The radius is half the length of the diameter:
[tex]\[ \text{Radius} = \frac{\text{Diameter}}{2} \][/tex]
So, using the diameter length:
[tex]\[ \text{Radius} = \frac{4.47213595499958}{2} \][/tex]
[tex]\[ = 2.23606797749979 \][/tex]
The radius is approximately [tex]\(2.236\)[/tex].
### Summary
Given the endpoints of the diameter [tex]\((3,6)\)[/tex] and [tex]\((-1,4)\)[/tex], we have:
- The center (midpoint) of the circle: [tex]\((1.0, 5.0)\)[/tex]
- The diameter length: [tex]\(4.47213595499958\)[/tex]
- The radius: [tex]\(2.23606797749979\)[/tex]
