Finding the Center of a Circle Given the Endpoints of Its Diameter

Geometry is a fundamental branch of mathematics that deals with shapes, sizes, and positions of figures. One common problem in geometry involves finding the center of a circle when the endpoints of its diameter are known. This article explains how to solve such a problem step-by-step, making it accessible to both students and professionals.

Understanding the Problem

A circle is a set of points in a plane that are all at the same distance from a given point, called the center. The diameter of a circle is any straight line segment that passes through the center and whose endpoints lie on the circle. The endpoints of the diameter divide the circle into two equal halves.

Key Concepts

To find the center of a circle given the endpoints of its diameter, one must locate the midpoint of the line segment connecting the endpoints. The midpoint is the point that divides the line segment into two equal parts. This midpoint is also the center of the circle.

Step-by-Step Solution

Given Information

The endpoints of the diameter are given as (A(7, 2)) and (B(-1, 8)).

Step 1: Identify Coordinates

Point A has coordinates (A(7, 2)) Point B has coordinates (B(-1, 8))

Step 2: Calculate the Midpoint

The midpoint of a line segment can be found using the midpoint formula:

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

Substitute the coordinates of points A and B into the formula:

[text{Midpoint} left(frac{7 (-1)}{2}, frac{2 8}{2}right)]

Step 3: Perform the Arithmetic

[text{Midpoint} left(frac{6}{2}, frac{10}{2}right) (3, 5)]

The midpoint of the line segment connecting points (A(7, 2)) and (B(-1, 8)) is ((3, 5)).

Conclusion

The center of a circle, given the endpoints of its diameter, is the midpoint of the line segment connecting these endpoints. In the case of endpoints (A(7, 2)) and (B(-1, 8)), the center of the circle is at ((3, 5)).

Practice Problems

1. If the endpoints of the diameter are (A(3, 4)) and (B(-5, 2)), find the center of the circle.

2. Given the endpoints of the diameter are (A(-2, -1)) and (B(4, 5)), determine the center of the circle.

Additional Resources

To further enhance your understanding and practice solving similar problems, refer to textbooks on geometry or online resources such as Khan Academy, which offer detailed explanations and interactive exercises.