Finding the Center of a Circle Given Its Diameter

This article explains how to find the center of a circle when given the endpoints of its diameter. The process involves using the properties of midpoints and coordinate geometry to derive the center's coordinates. Let's understand the mathematical steps involved and how to apply this knowledge in various scenarios.

Introduction to Circle Geometry

Circle geometry is a fundamental topic in mathematics. A circle is defined as the set of all points in a plane that are equidistant from a fixed point, known as the center. The line segment that passes through the center and touches the circle at two points is the diameter. Knowing the endpoints of the diameter can help us find the coordinates of the circle's center.

Understanding the Problem

Consider a scenario where the diameter of a circle touches the circle at two points, -48 and 1 - 2. Our task is to find the coordinates of the center of the circle.

Midpoint Formula Application

The midpoint formula, a key concept in coordinate geometry, finds the average of the coordinates of two points. For points ((x_1, y_1)) and ((x_2, y_2)), the midpoint is given by:

[(x_m, y_m) left(frac{x_1 x_2}{2}, frac{y_1 y_2}{2}right)]

In our problem, the diameter endpoints are (-48, -2) and (1, 8). By using these endpoints and the midpoint formula, we can find the coordinates of the center.

Calculating the Coordinates

To find the x-coordinate of the center, we use the formula for the x-coordinate of the midpoint:

[x_c frac{-48 1}{2} frac{-47}{2} -23.5]

Similarly, to find the y-coordinate of the center, we use the formula for the y-coordinate of the midpoint:

[y_c frac{-2 8}{2} frac{6}{2} 3]

Therefore, the coordinates of the center of the circle are ((-23.5, 3)).

Conclusion

In summary, to find the center of a circle given its diameter endpoints, use the midpoint formula to calculate the average of the x-coordinates and the average of the y-coordinates. This method applies to any circle, and by understanding the basic principles of coordinate geometry, you can confidently solve such problems.

For further reading and practice, consider exploring additional examples and exercises related to coordinate geometry and circle properties.