Circle Equation Through Square Vertices A Geometric Solution

Introduction

In the realm of analytical geometry, a fascinating problem arises when we're tasked with determining the equation of a circle that circumscribes a square, given the coordinates of two opposite vertices. This exploration delves into the intricacies of geometric relationships, coordinate geometry, and circle equations. Let's embark on a detailed journey to unravel this mathematical puzzle, which requires a blend of geometric intuition and algebraic manipulation. Understanding the properties of squares and circles is paramount, particularly the relationships between their centers, vertices, and radii. This problem not only sharpens our understanding of these geometric figures but also showcases the elegance with which coordinate geometry can be used to solve such problems.

Problem Statement

Consider a square where two opposite vertices are located at points (3, 1) and (-1, 3). The challenge lies in finding the equation of the circle that elegantly passes through all four vertices of this square. This requires a deep understanding of the geometric properties inherent in both squares and circles. The circle, by definition, is the locus of points equidistant from a central point, and the square, with its equal sides and right angles, imposes specific constraints on the circle's position and size. To solve this, we will leverage concepts such as the midpoint formula, the distance formula, and the standard equation of a circle.

Solution Approach

To solve this problem effectively, we need to follow a structured approach, breaking down the complex problem into manageable steps. The initial step involves finding the center of the circle, which coincides with the midpoint of the diagonal connecting the given vertices. This is a crucial step because the center serves as the reference point for defining the circle's position in the coordinate plane. Following this, we determine the radius of the circle, which is the distance from the center to any vertex of the square. This measurement is essential for defining the circle's size. Once we have both the center and the radius, we can plug these values into the standard equation of a circle, which provides a concise algebraic representation of the circle's geometry. This methodical approach ensures that we account for all the necessary geometric and algebraic considerations, leading to an accurate solution.

1. Finding the Center of the Circle

The center of the circle is a crucial element in defining its equation. For a circle circumscribing a square, the center of the circle coincides precisely with the midpoint of the diagonals of the square. Since the diagonals of a square bisect each other, finding the midpoint of the diagonal connecting the given vertices (3, 1) and (-1, 3) will reveal the center of our circle. The midpoint formula, a fundamental tool in coordinate geometry, states that the midpoint of a line segment with endpoints (x1, y1) and (x2, y2) is given by ((x1 + x2)/2, (y1 + y2)/2). Applying this formula to our vertices, we get the midpoint as ((3 + (-1))/2, (1 + 3)/2), which simplifies to (1, 2). Therefore, the center of the circle is located at the point (1, 2). This location serves as the anchor point around which the circle is drawn, and it is a critical parameter in the circle's equation.

2. Determining the Radius of the Circle

Having located the center of the circle, the next vital step is to determine its radius. The radius is the distance from the center of the circle to any point on its circumference, which in this case, is any vertex of the square. We can utilize the distance formula to calculate this radius. The distance formula, another cornerstone of coordinate geometry, states that the distance between two points (x1, y1) and (x2, y2) is given by √((x2 - x1)² + (y2 - y1)²). We can choose either of the given vertices, (3, 1) or (-1, 3), and the center (1, 2) to calculate the radius. Let's use the vertex (3, 1). Applying the distance formula, the radius r is √((3 - 1)² + (1 - 2)²) = √(2² + (-1)²) = √(4 + 1) = √5. Thus, the radius of the circle is √5 units. This measurement defines the size of the circle and is the final piece of information needed to write its equation.

3. Forming the Equation of the Circle

With the center and radius now determined, we can construct the equation of the circle. The standard equation of a circle with center (h, k) and radius r is given by (x - h)² + (y - k)² = r². In our case, the center (h, k) is (1, 2), and the radius r is √5. Substituting these values into the standard equation, we get (x - 1)² + (y - 2)² = (√5)², which simplifies to (x - 1)² + (y - 2)² = 5. This is the equation of the circle that passes through all four vertices of the square. This equation elegantly captures the geometric relationship between the points on the circle and its center, providing a concise algebraic representation of the circle's properties.

Final Answer

In conclusion, the equation of the circle that passes through all four vertices of the square, given the opposite vertices (3, 1) and (-1, 3), is (x - 1)² + (y - 2)² = 5. This result is achieved by systematically applying principles of coordinate geometry, such as the midpoint formula, the distance formula, and the standard equation of a circle. By first identifying the center of the circle as the midpoint of the diagonal and then calculating the radius as the distance from the center to a vertex, we were able to precisely define the circle's equation. This problem exemplifies the power of analytical geometry in solving geometric problems, providing a clear and concise algebraic representation of a geometric concept.

Keywords: circle equation, square vertices, analytical geometry, midpoint formula, distance formula, coordinate geometry.