Identifying Points On A Circle With Radius 5

Determining whether a point lies on a circle is a fundamental concept in geometry, and this article will explore how to identify points that reside on a circle centered at the origin with a radius of 5 units. We will employ the distance formula, a crucial tool in coordinate geometry, to calculate the distance between the origin and the given points. By comparing this distance to the radius of the circle, we can accurately ascertain which points lie on the circle. This exploration will not only solidify your understanding of circles and the distance formula but also enhance your problem-solving skills in analytical geometry.

The distance formula is the cornerstone of our approach, allowing us to precisely measure the distance between two points in a coordinate plane. The formula, √((x₂ - x₁)² + (y₂ - y₁)²), calculates the straight-line distance between points (x₁, y₁) and (x₂, y₂). In our scenario, one of the points is always the origin (0, 0), simplifying the distance calculation. To determine if a point lies on the circle, we will calculate the distance between the origin and the point in question. If this distance equals the radius of the circle (5 units in this case), then the point lies on the circle. If the distance is less than 5, the point lies inside the circle, and if it is greater than 5, the point lies outside the circle. This process enables us to methodically evaluate each option and pinpoint the point that satisfies the condition of being on the circle.

Understanding the equation of a circle centered at the origin is crucial for solving problems like this. The general equation of a circle centered at (0, 0) with a radius r is x² + y² = r². This equation is derived directly from the Pythagorean theorem and the distance formula. Any point (x, y) that satisfies this equation lies on the circle. In our case, the radius r is 5 units, so the equation of the circle is x² + y² = 5² or x² + y² = 25. To check if a point lies on the circle, we can substitute its coordinates into this equation. If the equation holds true, the point lies on the circle; otherwise, it does not. This method provides a quick and efficient way to verify our calculations using the distance formula. By understanding and applying the circle's equation, we can confidently confirm our results and deepen our understanding of the relationship between algebraic equations and geometric shapes.

Decoding the Options: Applying the Distance Formula

We are presented with four options: A. (2, √21), B. (2, √23), C. (2, 1), and D. (2, 3). To identify the point that lies on the circle, we will systematically apply the distance formula to each option. This involves calculating the distance between the origin (0, 0) and each given point and comparing the result to the radius of 5 units. By meticulously working through each option, we can confidently determine the correct answer. This process not only solves the problem at hand but also reinforces our understanding of how the distance formula applies in practical geometric scenarios.

Option A: (2, √21)

Let's begin by examining option A, the point (2, √21). Using the distance formula, we calculate the distance between (0, 0) and (2, √21) as follows:

Distance = √((2 - 0)² + (√21 - 0)²) = √(2² + (√21)²) = √(4 + 21) = √25 = 5

The distance calculated is exactly 5 units, which matches the radius of the circle. This indicates that the point (2, √21) lies precisely on the circle. However, to ensure we have the correct answer, we must also analyze the remaining options. This rigorous approach confirms our initial finding and solidifies our understanding of the problem-solving process. By thoroughly evaluating each option, we can be confident in our final conclusion.

Option B: (2, √23)

Next, we consider option B, the point (2, √23). Applying the distance formula, we find the distance between (0, 0) and (2, √23):

Distance = √((2 - 0)² + (√23 - 0)²) = √(2² + (√23)²) = √(4 + 23) = √27

The calculated distance is √27, which is greater than 5 (since √25 = 5). This means that the point (2, √23) lies outside the circle. Therefore, option B is not the correct answer. By methodically evaluating each option, we eliminate incorrect answers and narrow down the possibilities, bringing us closer to the correct solution. This step-by-step approach ensures accuracy and enhances our problem-solving skills.

Option C: (2, 1)

Now, let's analyze option C, the point (2, 1). Using the distance formula, we calculate the distance between (0, 0) and (2, 1):

Distance = √((2 - 0)² + (1 - 0)²) = √(2² + 1²) = √(4 + 1) = √5

The distance calculated is √5, which is approximately 2.24. This distance is less than the radius of 5 units, indicating that the point (2, 1) lies inside the circle. Therefore, option C is not the point we are looking for. By systematically applying the distance formula and comparing the results to the radius, we can accurately determine the location of each point relative to the circle.

Option D: (2, 3)

Finally, we evaluate option D, the point (2, 3). Applying the distance formula, we calculate the distance between (0, 0) and (2, 3):

Distance = √((2 - 0)² + (3 - 0)²) = √(2² + 3²) = √(4 + 9) = √13

The distance calculated is √13, which is approximately 3.61. This distance is also less than the radius of 5 units, meaning that the point (2, 3) lies inside the circle. Consequently, option D is not the correct answer. Our thorough analysis of each option allows us to confidently eliminate incorrect choices and identify the point that satisfies the given condition.

Conclusion: The Point on the Circle

After meticulously analyzing each option using the distance formula, we have determined that only point A, (2, √21), lies on the circle centered at the origin with a radius of 5 units. The distance between the origin and (2, √21) is exactly 5 units, confirming its location on the circle. Options B, C, and D were found to lie either outside or inside the circle. This exercise highlights the practical application of the distance formula in coordinate geometry and reinforces the understanding of circles and their properties. Understanding how to determine if a point lies on a circle is a fundamental skill in geometry, and this step-by-step approach provides a clear method for solving such problems.

Therefore, the correct answer is A. (2, √21).