91Ó°ÊÓ

When dealing with lines, the midpoint formula is a handy tool. It's particularly useful for finding the center of a circle if you know the endpoints of a diameter. The formula for the midpoint is \( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \). You simply take the average of the x-coordinates and the y-coordinates separately.
In our exercise, the endpoints of the diameter are \((4, 2)\) and \((-3, 5)\). By plugging these into the formula, we get:

• \( x \)=\( \frac{4 + (-3)}{2} = \frac{1}{2} \)
• \( y \)=\( \frac{2 + 5}{2} = \frac{7}{2} \)

Hence, the midpoint, which serves as the center of the circle, is \( \left( \frac{1}{2}, \frac{7}{2} \right) \). Using the midpoint formula allows us to locate the center of the circle easily, using simple arithmetic.

To calculate the length of a line segment, the distance formula is your go-to tool. This formula is especially helpful in determining the diameter of the circle if you have its endpoints. The formula is given by \( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).
Let's apply this formula to the endpoints (4, 2) and (-3, 5) from our problem.

• First calculate the change in x: \(-3 - 4 = -7\)
• Then the change in y: \(5 - 2 = 3\)
• Substitute these values into the formula: \( \sqrt{(-7)^2 + 3^2} = \sqrt{49 + 9} = \sqrt{58} \)

Thus, the distance between the two points, which is the diameter of the circle, is \( \sqrt{58} \). The distance formula gives us not just a number but the exact distance on a coordinate plane.

Understanding the relationship between diameter and radius is key in circle geometry. The diameter is the longest line that can fit across a circle, passing through the center. The radius, on the other hand, is half the diameter.
Knowing the diameter, as found with the distance formula \( \sqrt{58} \), allows us to calculate the radius:

• Simply divide the diameter by 2: \( \frac{\sqrt{58}}{2} \)

Using this relationship helps us in finding critical circle information, like its equation. The radius is essential since it directly appears in the circle equation formula, helping represent the circle in algebraic form.

The center of a circle is a fundamental aspect when writing its equation. With the center found using the midpoint formula, \( \left( \frac{1}{2}, \frac{7}{2} \right) \), we can set up the circle's equation. This center, made up of coordinates (h, k), defines the circle's position on a plane.
Substituting the center into the general circle equation format \((x - h)^2 + (y - k)^2 = r^2\), connects its geometrical properties with algebra:

• Substitute \(h = \frac{1}{2}\), and \(k = \frac{7}{2}\), giving \((x - \frac{1}{2})^2 + (y - \frac{7}{2})^2 = r^2\)

The center, though just a pair of coordinates, becomes instrumental in fully illustrating the circle's attributes when combined with radius information. Understanding this concept allows you to interpret and construct the mathematical form of a circle.