Finding The Radius Of A Circle From Its Equation (x-7)² + (y-10)² = 4

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In the realm of geometry, circles hold a fundamental position, captivating mathematicians and enthusiasts alike with their elegant symmetry and ubiquitous presence in the world around us. From the celestial dance of planets to the intricate designs of snowflakes, circles manifest in countless forms, making their study an essential pursuit. At the heart of understanding circles lies the concept of their radius, a crucial parameter that defines their size and characteristics.

Decoding the Circle Equation: (x-7)² + (y-10)² = 4

When confronted with the equation of a circle, such as (x-7)² + (y-10)² = 4, the immediate goal is to decipher its meaning and extract the key information it holds. This equation is presented in the standard form of a circle's equation, a format that readily reveals the circle's center and radius. The standard form equation is expressed as:

(x - h)² + (y - k)² = r²

where:

  • (h, k) represents the coordinates of the center of the circle.
  • r denotes the radius of the circle.

By meticulously comparing the given equation, (x-7)² + (y-10)² = 4, with the standard form, we can embark on a journey to unveil the circle's hidden attributes. Through careful observation, we can discern that the center of the circle is located at the point (7, 10). This signifies that the circle is positioned on the Cartesian plane with its central point anchored at these coordinates.

Furthermore, the right-hand side of the equation, 4, corresponds to r², the square of the radius. To determine the actual radius, we must embark on a mathematical quest to find the square root of 4. The square root of 4 is 2, which unveils the radius of the circle as 2 units. This signifies that the distance from the center of the circle to any point on its circumference is precisely 2 units.

Therefore, by meticulously dissecting the equation (x-7)² + (y-10)² = 4, we have successfully extracted the circle's vital statistics: its center at (7, 10) and its radius of 2 units. This knowledge empowers us to visualize and analyze the circle's position and size within the coordinate plane.

The Significance of the Radius: A Circle's Defining Feature

The radius of a circle is more than just a numerical value; it serves as a fundamental building block that dictates the circle's very essence. This seemingly simple measurement holds the key to unlocking a wealth of information about the circle's properties and its relationship to the world around it.

  • Size and Scale: The radius is the ultimate determinant of a circle's size. A larger radius translates to a larger circle, encompassing a greater area and stretching further across the plane. Conversely, a smaller radius defines a more compact circle, nestled closer to its center.

  • Circumference and Area: The radius plays a pivotal role in calculating a circle's circumference and area, two essential measurements that quantify its dimensions. The circumference, the distance around the circle, is directly proportional to the radius, expressed by the formula:

    Circumference = 2πr

    where π (pi) is a mathematical constant approximately equal to 3.14159. Similarly, the area enclosed by the circle is intimately linked to the radius, governed by the formula:

    Area = πr²

    These formulas underscore the radius's fundamental influence on a circle's size and extent.

  • Geometric Relationships: The radius forms the cornerstone of numerous geometric relationships involving circles. It defines the distance from the center to any point on the circumference, acting as the thread that connects the center to the circle's outer limits. Moreover, the radius is instrumental in constructing tangents, lines that graze the circle at a single point, and chords, line segments that connect two points on the circumference. These geometric constructs further highlight the radius's central role in shaping a circle's form and its interactions with other geometric entities.

In essence, the radius is the lifeblood of a circle, the defining characteristic that dictates its size, proportions, and its place within the geometric landscape. Understanding the radius is paramount to grasping the essence of a circle and its multifaceted applications.

Applying the Knowledge: Solving the Problem

Now, armed with a comprehensive understanding of circle equations and the significance of the radius, let us return to the original question: What is the radius of a circle whose equation is (x-7)² + (y-10)² = 4?

We have already meticulously dissected the equation and identified its key components. By comparing it to the standard form equation, (x - h)² + (y - k)² = r², we have established that the center of the circle is at (7, 10) and that r² = 4.

To find the radius, we simply take the square root of 4, which yields 2. Therefore, the radius of the circle is 2 units.

This answer aligns perfectly with option A, which states that the radius is 2 units. Thus, option A is the correct solution to the problem.

Conclusion: The Power of Understanding

In conclusion, we have embarked on a journey to unravel the mystery of the radius within the context of a circle's equation. By meticulously dissecting the equation (x-7)² + (y-10)² = 4, we have successfully identified the circle's center and, more importantly, its radius.

We have also delved into the profound significance of the radius, recognizing its role as the defining feature of a circle, influencing its size, circumference, area, and its interactions with other geometric elements. Understanding the radius is not merely about solving equations; it is about grasping the essence of a fundamental geometric shape and its pervasive presence in the world around us.

This exploration underscores the power of mathematical understanding. By mastering the concepts and techniques associated with circles and their equations, we empower ourselves to decipher geometric puzzles, solve real-world problems, and appreciate the elegance and order that mathematics brings to our perception of the universe.