Finding The Radius Of A Circle Given Its Equation (x+7)^2+(y+7)^2=16

In the realm of mathematics, understanding the equation of a circle is fundamental in analytic geometry. The standard equation of a circle provides a concise way to represent a circle's properties, including its center and radius. The equation is generally expressed as (x - h)² + (y - k)² = r², where (h, k) represents the coordinates of the center of the circle, and r signifies the radius. This formula stems directly from the Pythagorean theorem, applied to any point on the circle and the circle's center. Mastering this equation opens doors to solving various geometric problems, from finding distances to understanding the relationships between circles and other geometric shapes.

Delving deeper into the equation, we can see how each component contributes to defining the circle. The values h and k, when subtracted from x and y respectively, shift the circle's center away from the origin (0, 0). If h and k are both zero, the circle is centered at the origin. The r² term on the right side of the equation is particularly crucial as it directly relates to the circle's size. Taking the square root of r² gives us the radius, which is the distance from the center of the circle to any point on its circumference. This simple yet powerful equation allows mathematicians and students alike to quickly ascertain key characteristics of a circle given its algebraic representation.

Furthermore, the equation (x - h)² + (y - k)² = r² is not just an abstract formula; it has practical applications in various fields. In computer graphics, for example, it's used to draw circles and arcs. In physics, it can describe the path of an object moving in circular motion. In engineering, it's used in the design of circular components and structures. Therefore, a strong grasp of this equation is beneficial across numerous disciplines. The beauty of this equation lies in its simplicity and versatility, encapsulating the essence of a circle in a compact mathematical form. By understanding how changes in h, k, and r affect the circle, one can easily manipulate and analyze circular shapes in a variety of contexts.

Analyzing the Given Equation (x+7)^2+(y+7)2=16

In this particular problem, we are presented with the equation of a circle: (x + 7)² + (y + 7)² = 16. To decipher the radius, we need to map this equation onto the standard form (x - h)² + (y - k)² = r². The first step is to recognize that the plus signs within the parentheses can be rewritten as subtractions of negative numbers. Thus, (x + 7)² is equivalent to (x - (-7))², and (y + 7)² is equivalent to (y - (-7))². This transformation allows us to identify the center of the circle as (-7, -7).

Now, focusing on the right side of the equation, we have 16. Recall that in the standard form, the right side represents r², where r is the radius. Therefore, 16 is equal to r². To find the radius, we need to calculate the square root of 16. The square root of 16 is 4, since 4 * 4 = 16. This tells us that the radius of the circle is 4 units. It's crucial to remember that the radius is a distance, and thus it must be a positive value. Taking the square root of a number yields both a positive and a negative solution, but in this context, we only consider the positive root.

Understanding this process is vital for handling similar problems. The ability to quickly convert an equation into its standard form and extract the relevant information, such as the center and radius, is a key skill in coordinate geometry. Moreover, this skill extends to more complex problems involving circles, such as finding tangent lines, determining intersections with other shapes, and solving geometric proofs. By breaking down the equation into its components and relating them to the geometric properties of the circle, we can effectively solve a wide range of mathematical challenges. In this instance, the correct identification of the radius as 4 demonstrates a clear understanding of the circle equation and its implications.

Selecting the Correct Answer

Having determined that the radius of circle O is 4, we now need to select the corresponding answer from the provided options. The options are: A. 3, B. 4, C. 7, D. 9, and E. 16. Our calculation clearly shows that the radius is 4, which directly corresponds to option B. Therefore, the correct answer is B. 4. This step-by-step process of analyzing the equation and identifying the radius underscores the importance of careful calculation and logical deduction in mathematics.

It's essential to differentiate between the radius (r) and the r² term in the equation. A common mistake is to select 16 (option E) as the radius, but it's crucial to remember that 16 represents the square of the radius. This highlights the need for a thorough understanding of the formula and the relationships between its components. By correctly identifying 4 as the radius, we demonstrate a solid grasp of the circle equation and its application. This careful approach ensures accuracy and prevents errors that can arise from misinterpreting the equation.

Furthermore, understanding the significance of each option helps reinforce the concept. Option A (3) and Options C (7) and D (9) are incorrect because they do not match the calculated radius of 4. Option E (16) represents the square of the radius, not the radius itself. By recognizing the correct answer and understanding why the other options are incorrect, we solidify our understanding of the underlying mathematical principles. This thorough approach to problem-solving is crucial not only for answering specific questions but also for developing a deeper understanding of mathematics as a whole.

Therefore, the final answer, selected with confidence after careful analysis, is B. 4. This demonstrates a clear understanding of the circle equation, the process of extracting the radius, and the importance of accurate calculation and interpretation.

Conclusion

In summary, determining the radius of a circle from its equation is a straightforward process when the standard form (x - h)² + (y - k)² = r² is understood. By correctly identifying the r² term and taking its square root, we can find the radius. In the given equation, (x + 7)² + (y + 7)² = 16, the radius was calculated to be 4. This underscores the importance of understanding mathematical formulas and their applications in solving problems.

This exercise also highlights the significance of careful analysis and attention to detail. Misinterpreting the equation or overlooking the square root operation can lead to incorrect answers. The ability to break down a problem into smaller, manageable steps is crucial for success in mathematics. By understanding the individual components of the equation and their relationships, we can confidently arrive at the correct solution.

Finally, the ability to apply this knowledge to similar problems is a key outcome of understanding the concept. Whether encountering different circle equations or more complex geometric problems, the underlying principles remain the same. A strong foundation in the fundamentals allows for adaptability and problem-solving proficiency in various mathematical contexts. Thus, mastering the circle equation is not just about solving one problem, but about building a broader understanding of mathematical concepts and their applications.