Finding Points On A Circle With Radius 5 And Center (6,1)

ADMIN · · 3 min read

Table Of Content

  1. Problem Statement
  2. Understanding the Distance Formula
  3. Applying the Distance Formula to Each Point
    1. Point Q(1,11)
    2. Point R(2,4)
    3. Point S(4,-4)
    4. Point T(9,-2)
  4. Solution and Conclusion
  5. Key Concepts and Takeaways
  6. Practice Problems
  7. Further Exploration

In this article, we will delve into the process of identifying which point lies on a circle with a given radius and center. This is a fundamental concept in coordinate geometry, often encountered in mathematics courses and standardized tests. We will analyze the given problem step by step, providing a comprehensive explanation and solution. Understanding how to determine if a point lies on a circle is crucial for various applications in geometry, physics, and engineering. This article aims to provide a clear and concise method for solving such problems, ensuring that readers can confidently tackle similar questions in the future.

Problem Statement

The problem presents us with a circle that has a radius of 5 units and a center located at the point P(6,1). Our task is to determine which of the given points, namely Q(1,11), R(2,4), S(4,-4), and T(9,-2), lies on this circle. To solve this, we will use the distance formula, which is derived from the Pythagorean theorem. The distance formula allows us to calculate the distance between two points in a coordinate plane. If the distance between a given point and the center of the circle is equal to the radius, then that point lies on the circle. This method is a direct application of the definition of a circle: the set of all points equidistant from a central point.

Understanding the Distance Formula

The distance formula is a cornerstone of coordinate geometry. It enables us to calculate the distance between any two points in a coordinate plane. The formula is derived from the Pythagorean theorem, which relates the sides of a right triangle. Given two points, $(x_1, y_1)$ and $(x_2, y_2)$, the distance d between them is given by:

d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

This formula calculates the length of the hypotenuse of a right triangle whose legs are the differences in the x-coordinates and the y-coordinates of the two points. In the context of our problem, we will use this formula to calculate the distance between each of the given points (Q, R, S, and T) and the center of the circle, P(6,1). If the calculated distance is equal to the radius of the circle (5 units), then the point lies on the circle. Mastering the distance formula is essential for solving a wide range of geometric problems, including those involving circles, lines, and other shapes in the coordinate plane. The formula provides a direct and efficient way to determine the spatial relationship between points, making it a fundamental tool in mathematical analysis.

Applying the Distance Formula to Each Point

To determine which point lies on the circle, we will apply the distance formula to each of the given points (Q, R, S, and T) with respect to the center P(6,1). We will calculate the distance between each point and the center, and if the distance equals the radius (5 units), we will conclude that the point lies on the circle. This process involves substituting the coordinates of the points into the distance formula and performing the calculations. Let's go through each point step by step:

Point Q(1,11)

The distance between P(6,1) and Q(1,11) is:

d=(1−6)2+(11−1)2=(−5)2+(10)2=25+100=125=55d = \sqrt{(1 - 6)^2 + (11 - 1)^2} = \sqrt{(-5)^2 + (10)^2} = \sqrt{25 + 100} = \sqrt{125} = 5\sqrt{5}

Since $5\sqrt{5}$ is not equal to 5, point Q does not lie on the circle.

Point R(2,4)

The distance between P(6,1) and R(2,4) is:

d=(2−6)2+(4−1)2=(−4)2+(3)2=16+9=25=5d = \sqrt{(2 - 6)^2 + (4 - 1)^2} = \sqrt{(-4)^2 + (3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5

Since the distance is 5, which is the radius of the circle, point R lies on the circle.

Point S(4,-4)

The distance between P(6,1) and S(4,-4) is:

d=(4−6)2+(−4−1)2=(−2)2+(−5)2=4+25=29d = \sqrt{(4 - 6)^2 + (-4 - 1)^2} = \sqrt{(-2)^2 + (-5)^2} = \sqrt{4 + 25} = \sqrt{29}

Since $\sqrt{29}$ is not equal to 5, point S does not lie on the circle.

Point T(9,-2)

The distance between P(6,1) and T(9,-2) is:

d=(9−6)2+(−2−1)2=(3)2+(−3)2=9+9=18=32d = \sqrt{(9 - 6)^2 + (-2 - 1)^2} = \sqrt{(3)^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2}

Since $3\sqrt{2}$ is not equal to 5, point T does not lie on the circle.

Solution and Conclusion

After applying the distance formula to each point, we found that only point R(2,4) has a distance of 5 units from the center P(6,1). This distance is equal to the radius of the circle, confirming that point R lies on the circle. Therefore, the correct answer is B. This problem demonstrates the practical application of the distance formula in determining the relationship between points and circles in a coordinate plane. The distance formula is a powerful tool that allows us to analyze geometric properties and solve problems involving distances and positions of points. In summary, by calculating the distances between the given points and the center of the circle, we were able to definitively identify the point that lies on the circle. This method is not only applicable to this specific problem but also serves as a general approach for solving similar problems in coordinate geometry.

Therefore, the correct answer is B. R(2,4).

Key Concepts and Takeaways

In this detailed solution, several key concepts and takeaways have emerged that are crucial for understanding and solving similar problems in coordinate geometry. Understanding these concepts will enhance your ability to tackle a wider range of geometric challenges and improve your problem-solving skills. Here are some of the essential points to remember:

  1. Distance Formula: The distance formula is a fundamental tool for calculating the distance between two points in a coordinate plane. It is derived from the Pythagorean theorem and is expressed as $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. Mastering this formula is essential for solving problems involving distances and positions of points.

  2. Definition of a Circle: A circle is defined as the set of all points equidistant from a central point. This distance is known as the radius of the circle. Understanding this definition is key to determining whether a point lies on a circle.

  3. Applying the Distance Formula to Circles: To determine if a point lies on a circle, calculate the distance between the point and the center of the circle using the distance formula. If this distance is equal to the radius, then the point lies on the circle. This method provides a direct and efficient way to solve such problems.

  4. Step-by-Step Approach: When solving geometric problems, it is often helpful to break down the problem into smaller, manageable steps. In this case, we applied the distance formula to each point individually, making the process clear and organized. A methodical approach can help prevent errors and ensure accuracy.

  5. Geometric Interpretation: Visualizing the problem geometrically can aid in understanding the solution. In this case, picturing the circle and the points in the coordinate plane can provide a clearer understanding of the problem and the relationship between the points and the circle.

By understanding and applying these key concepts, you will be well-equipped to solve a variety of problems involving circles, distances, and coordinate geometry. The ability to apply these concepts effectively is crucial for success in mathematics and related fields.

Practice Problems

To reinforce your understanding of the concepts discussed in this article, it's essential to practice solving similar problems. Practice helps solidify your grasp of the distance formula and its application in determining points on a circle. Here are a few practice problems that you can try:

  1. Problem 1: A circle has a center at (2, -3) and a radius of 4 units. Which of the following points lies on the circle?

    • A. (2, 1)
    • B. (6, -3)
    • C. (2, -7)
    • D. (5, -3)

  2. Problem 2: Determine whether the point (-1, 2) lies on a circle with a center at (3, 1) and a radius of $\sqrt{17}$ units.

  3. Problem 3: A circle has a center at (-4, 5) and a radius of 6 units. Find the equation of the circle and determine if the point (2, 5) lies on the circle.

  4. Problem 4: If the points (0, 0) and (4, 0) are endpoints of the diameter of a circle, what is the center and radius of the circle? Also, determine if the point (2, 2) lies on the circle.

Solving these practice problems will help you become more comfortable with the distance formula and its applications. It will also enhance your problem-solving skills and your ability to approach geometric problems with confidence. Make sure to follow the step-by-step approach we used in the detailed solution and apply the key concepts discussed in this article. Remember, practice is key to mastering any mathematical concept.

By working through these problems, you will not only reinforce your understanding of the distance formula and its application to circles but also develop a deeper appreciation for the beauty and logic of coordinate geometry. Each problem provides an opportunity to apply the concepts in a different context, helping you build a versatile and robust skill set. So, take the time to practice, and you will find yourself becoming more proficient and confident in your ability to solve geometric problems.

Further Exploration

To further enhance your understanding and skills in coordinate geometry, there are several avenues you can explore. Delving deeper into related topics and concepts will not only broaden your knowledge but also provide a more comprehensive perspective on the subject. Here are some areas you might consider for further exploration:

  1. Equation of a Circle: Understanding the equation of a circle is crucial for solving more complex problems. The standard equation of a circle with center (h, k) and radius r is given by $(x - h)^2 + (y - k)^2 = r^2$. Learning how to derive and use this equation will allow you to solve a variety of problems related to circles.

  2. Geometric Transformations: Explore geometric transformations such as translations, rotations, reflections, and dilations. Understanding how these transformations affect points and shapes in the coordinate plane can provide valuable insights and problem-solving techniques.

  3. Conic Sections: Study other conic sections such as ellipses, parabolas, and hyperbolas. These shapes are closely related to circles and share many similar properties. Understanding conic sections will broaden your knowledge of geometric shapes and their equations.

  4. Applications of Coordinate Geometry: Investigate the applications of coordinate geometry in various fields such as physics, engineering, and computer graphics. Understanding how these concepts are used in real-world scenarios can provide a deeper appreciation for the subject.

  5. Advanced Problem-Solving Techniques: Explore advanced problem-solving techniques such as using systems of equations, inequalities, and trigonometric functions to solve geometric problems. These techniques can be particularly useful for tackling more challenging problems.

By exploring these areas, you will not only expand your knowledge of coordinate geometry but also develop a more versatile and comprehensive skill set. The journey of learning mathematics is continuous, and each new concept you learn builds upon the previous ones, creating a strong foundation for future studies.

This article has provided a thorough explanation of how to determine if a point lies on a circle, along with practice problems and suggestions for further exploration. By mastering these concepts and continuing your learning journey, you will be well-equipped to tackle a wide range of geometric challenges.

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