91Ó°ÊÓ

The midpoint formula is a handy tool in geometry that helps us find the center of a line segment. Given two points, the formula finds their average and provides the point directly in the middle. If you have points \(P(x_1, y_1)\) and \(Q(x_2, y_2)\), the midpoint \(M\) is calculated as:

• \(\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\)

In the context of a circle, especially when dealing with a diameter, this point becomes significant. The midpoint of a diameter serves as the circle's center.
For example, if the endpoints of a diameter are \((-1, 3)\) and \((7, -5)\), computing the midpoint gives us the center at \((3, -1)\). This practical application of the midpoint formula allows us to easily locate the center, an essential step in defining a circle's equation.

The distance formula is used to determine how far apart two points are in a plane. This is crucial in calculating the length of the diameter of a circle, which then helps in finding the circle's radius. Derived from the Pythagorean theorem, the distance between two points \(P(x_1, y_1)\) and \(Q(x_2, y_2)\) is:

• \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

Applying this to points \((-1,3)\) and \((7,-5)\), we find the distance to be \(8\sqrt{2}\). This length represents the diameter of the circle.
To find the radius, simply halve this distance, which provides \(4\sqrt{2}\). The distance formula thus provides the necessary value of the radius, allowing us to express other circle properties.

The general equation of a circle ties all information about the circle into one neat expression. A circle with center \((h, k)\) and radius \(r\) is modeled by:

• \((x - h)^2 + (y - k)^2 = r^2\)

This concise equation embodies the circle’s defining characteristics.
The center \((3, -1)\) and radius \(4\sqrt{2}\) yield the equation \((x - 3)^2 + (y + 1)^2 = 32\). Here, the transformations \((x - h)\) and \((y - k)\) adjust the general form to match the center location while \(r^2\) adjusts the size of the circle.
Understanding this equation is an essential skill. It allows you to effectively describe the size and location of a circle on the coordinate plane.