Which Point Is On The Circle Centered At The Origin With A Radius Of 5 Units?Distance Formula: ( X 2 − X 1 ) 2 + ( Y 2 − Y 1 ) 2 \sqrt{(x_2-x_1)^2+(y_2-y_1)^2} ( X 2 − X 1 ) 2 + ( Y 2 − Y 1 ) 2 A. (2, { \sqrt{21}$}$) B. (2, { \sqrt{23}$}$) C. (2, 1) D. (2, 3)
Introduction
In mathematics, a circle is a set of points that are all equidistant from a central point called the center. The distance between the center and any point on the circle is called the radius. In this article, we will explore the concept of a circle centered at the origin with a radius of 5 units and determine which point is on the circle.
Understanding the Circle
A circle centered at the origin with a radius of 5 units can be represented by the equation:
x^2 + y^2 = 5^2
This equation represents all the points that are 5 units away from the origin. To find the points on the circle, we need to find the values of x and y that satisfy this equation.
Using the Distance Formula
The distance formula is a mathematical formula that calculates the distance between two points in a coordinate plane. The formula is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)
In this case, we want to find the distance between the origin (0, 0) and the point (x, y). We can plug in the values into the formula and simplify:
d = √((x - 0)^2 + (y - 0)^2) d = √(x^2 + y^2)
Since the point (x, y) is on the circle, the distance d is equal to the radius, which is 5 units. Therefore, we can set up the equation:
√(x^2 + y^2) = 5
Solving for x and y
To solve for x and y, we can square both sides of the equation:
x^2 + y^2 = 5^2
x^2 + y^2 = 25
Now, we can expand the equation:
x^2 + y^2 = 25
This is the equation of a circle centered at the origin with a radius of 5 units. To find the points on the circle, we need to find the values of x and y that satisfy this equation.
Analyzing the Options
We are given four options:
A. (2, √21) B. (2, √23) C. (2, 1) D. (2, 3)
We can plug in the values of x and y into the equation x^2 + y^2 = 25 and check if they satisfy the equation.
Checking Option A
For option A, x = 2 and y = √21. Plugging in these values into the equation, we get:
(2)^2 + (√21)^2 = 4 + 21 = 25
This satisfies the equation, so option A is a possible solution.
Checking Option B
For option B, x = 2 and y = √23. Plugging in these values into the equation, we get:
(2)^2 + (√23)^2 = 4 + 23 = 27
This does not satisfy the equation, so option B is not a possible solution.
Checking Option C
For option C, x = 2 and y = 1. Plugging in these values into the equation, we get:
(2)^2 + (1)^2 = 4 + 1 = 5
This does not satisfy the equation, so option C is not a possible solution.
Checking Option D
For option D, x = 2 and y = 3. Plugging in these values into the equation, we get:
(2)^2 + (3)^2 = 4 + 9 = 13
This does not satisfy the equation, so option D is not a possible solution.
Conclusion
Based on our analysis, we can conclude that option A, (2, √21), is the only point that satisfies the equation x^2 + y^2 = 25. Therefore, the correct answer is:
Introduction
In our previous article, we explored the concept of a circle centered at the origin with a radius of 5 units and determined which point is on the circle. In this article, we will answer some frequently asked questions related to this topic.
Q: What is the equation of a circle centered at the origin with a radius of 5 units?
A: The equation of a circle centered at the origin with a radius of 5 units is x^2 + y^2 = 5^2, which simplifies to x^2 + y^2 = 25.
Q: How do I find the points on the circle?
A: To find the points on the circle, you need to find the values of x and y that satisfy the equation x^2 + y^2 = 25. You can use the distance formula to find the distance between the origin and the point (x, y), and then set it equal to the radius, which is 5 units.
Q: What is the distance formula?
A: The distance formula is a mathematical formula that calculates the distance between two points in a coordinate plane. The formula is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)
Q: How do I use the distance formula to find the points on the circle?
A: To use the distance formula to find the points on the circle, you need to plug in the values of x and y into the formula and simplify. Then, you can set the distance d equal to the radius, which is 5 units, and solve for x and y.
Q: What are some common mistakes to avoid when working with circles?
A: Some common mistakes to avoid when working with circles include:
- Not using the correct equation for the circle
- Not plugging in the correct values into the distance formula
- Not simplifying the equation correctly
- Not checking the solutions to make sure they satisfy the equation
Q: How do I check if a point is on the circle?
A: To check if a point is on the circle, you need to plug in the values of x and y into the equation x^2 + y^2 = 25 and check if it satisfies the equation. If it does, then the point is on the circle.
Q: What are some real-world applications of circles?
A: Circles have many real-world applications, including:
- Designing circular shapes for buildings and bridges
- Creating circular patterns for art and design
- Calculating distances and angles in navigation and surveying
- Modeling the motion of objects in physics and engineering
Conclusion
In this article, we answered some frequently asked questions related to the circle centered at the origin with a radius of 5 units. We hope that this article has been helpful in understanding this topic and has provided you with a better understanding of circles and their applications.
Additional Resources
If you want to learn more about circles and their applications, here are some additional resources that you may find helpful:
- Online tutorials and videos on geometry and trigonometry
- Math textbooks and workbooks on geometry and trigonometry
- Online forums and communities for math enthusiasts
- Math apps and software for practicing geometry and trigonometry problems
Final Thoughts
We hope that this article has been helpful in understanding the circle centered at the origin with a radius of 5 units. Remember to always check your solutions and to use the correct equation for the circle. With practice and patience, you will become proficient in working with circles and their applications.