Finding Circle B's Center A Step-by-Step Solution

In the realm of geometry, circles hold a fundamental place, and understanding their properties is crucial for solving various problems. One such property is the center of a circle, which serves as the anchor point around which the circle's circumference is drawn. In this article, we will delve into a specific scenario involving circle B, where its radius is 11 units and its center is located 4 units above the origin along the y-axis. Our primary goal is to pinpoint the precise coordinates of the center of circle B. This exploration will not only enhance our understanding of circles but also demonstrate the application of coordinate geometry principles.

In coordinate geometry, the center of a circle plays a pivotal role in defining its position on the Cartesian plane. The center, represented as a coordinate pair (x, y), serves as the focal point from which all points on the circle's circumference are equidistant. This distance, known as the radius, further characterizes the circle's size. To effectively determine the center of circle B, we must carefully analyze the provided information and translate it into coordinate form. In this section, we will meticulously dissect the given clues to unveil the coordinates of circle B's center.

Understanding the Problem Statement

To accurately determine the center of circle B, we must first dissect the given information. We are told that the circle's center is situated 4 units above the origin along the y-axis. This seemingly simple statement holds the key to unlocking the circle's center coordinates. The term "origin" refers to the point (0, 0) on the Cartesian plane, the intersection of the x-axis and y-axis. Moving "above the origin" implies a vertical displacement along the positive y-axis. Therefore, a point located 4 units above the origin will have a y-coordinate of 4. The x-coordinate, however, remains unchanged since there is no horizontal displacement. This meticulous interpretation of the problem statement sets the stage for identifying the correct center coordinates.

Identifying the Center Coordinates

Building upon our understanding of the problem statement, we can now pinpoint the coordinates of circle B's center. As established, the circle's center lies 4 units above the origin along the y-axis. This translates directly to a y-coordinate of 4. Since there is no horizontal displacement from the origin, the x-coordinate remains at 0. Consequently, the center of circle B is located at the point (0, 4). This conclusion aligns perfectly with the given information and provides a clear understanding of the circle's position on the Cartesian plane. Visualizing this point on the coordinate plane further solidifies our understanding of its location relative to the origin and axes.

To reinforce our understanding and demonstrate the problem-solving process, let's analyze the given options: (A) (0,0), (B) (-4,0), (C) (4,0), and (D) (0,4). By systematically evaluating each option, we can confirm our solution and eliminate any potential misconceptions.

Option A: (0, 0)

Option A, (0, 0), represents the origin itself. While the origin serves as a reference point in our problem, it does not satisfy the condition that the circle's center is located 4 units above the origin along the y-axis. Therefore, option A can be confidently eliminated. This analysis highlights the importance of adhering to the specific conditions outlined in the problem statement.

Option B: (-4, 0)

Option B, (-4, 0), represents a point located 4 units to the left of the origin along the x-axis. This contradicts the given information, which specifies that the circle's center is positioned above the origin along the y-axis. Thus, option B is not the correct answer. This analysis reinforces the significance of accurately interpreting directional cues within the problem statement.

Option C: (4, 0)

Option C, (4, 0), represents a point located 4 units to the right of the origin along the x-axis. This point, like option B, deviates from the specified condition of being 4 units above the origin along the y-axis. Consequently, option C is also an incorrect choice. This analysis underscores the importance of distinguishing between horizontal and vertical displacements when determining coordinates.

Option D: (0, 4)

Option D, (0, 4), precisely matches our derived coordinates for the center of circle B. This point lies 4 units above the origin along the y-axis, fulfilling the given condition. Therefore, option D is the correct answer. This confirmation reinforces the accuracy of our problem-solving approach and validates our understanding of coordinate geometry principles.

In conclusion, by carefully analyzing the given information and applying principles of coordinate geometry, we have successfully determined that the center of circle B is located at the point (0, 4). This solution aligns perfectly with the condition that the center is 4 units above the origin along the y-axis. Our analysis of the provided options further solidified this conclusion, reinforcing the importance of systematic problem-solving and attention to detail. Understanding the properties of circles, such as the center and radius, is crucial for tackling more complex geometric challenges.

To deepen your understanding of circles and coordinate geometry, consider exploring the following topics:

• Equation of a Circle: Learn how to represent a circle algebraically using its center and radius.
• Distance Formula: Apply the distance formula to calculate the distance between points on a circle and its center.
• Circle Theorems: Investigate various theorems related to circles, such as the inscribed angle theorem and the tangent-chord theorem.
• Applications of Circles: Discover real-world applications of circles in fields like engineering, architecture, and physics.

By delving into these areas, you can further enhance your problem-solving skills and gain a more comprehensive understanding of the fascinating world of geometry.