The Circle Centered at the Origin

Oct 13, 2025 by recoveryshake.com 116 views

Introduction

In this article, we will explore the concept of circles and their properties. Specifically, we will focus on finding a point that lies on a circle centered at the origin with a radius of 5 units. To do this, we will use the distance formula, which is a fundamental concept in mathematics.

The Distance Formula

The distance formula is a mathematical formula that calculates the distance between two points in a coordinate plane. It is given by:

$\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}$

This formula is used to find the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ in a coordinate plane.

The Circle Centered at the Origin

A circle is a set of points that are equidistant from a central point called the center. In this case, the circle is centered at the origin, which is the point $(0, 0)$ in the coordinate plane. The radius of the circle is 5 units, which means that every point on the circle is 5 units away from the origin.

Finding the Point on the Circle

To find the point on the circle, we need to use the distance formula. We know that the point lies on the circle, so the distance from the origin to the point is equal to the radius of the circle, which is 5 units. We can use the distance formula to set up an equation and solve for the point.

Let's consider the options given:

A. $(2, \sqrt{21})$ B. $(2, \sqrt{23})$ C. $(2, 1)$ D. $(2, \sqrt{25})$

We can use the distance formula to calculate the distance from the origin to each of these points and see which one is equal to 5 units.

Calculating the Distance

Let's calculate the distance from the origin to each of the points:

A. $(2, \sqrt{21})$

Using the distance formula, we get:

$\sqrt{\left(2-0\right)^2+\left(\sqrt{21}-0\right)^2}$ $=\sqrt{4+21}$ $=\sqrt{25}$ $=5$

B. $(2, \sqrt{23})$

Using the distance formula, we get:

$\sqrt{\left(2-0\right)^2+\left(\sqrt{23}-0\right)^2}$ $=\sqrt{4+23}$ $=\sqrt{27}$ $\neq 5$

C. $(2, 1)$

Using the distance formula, we get:

$\sqrt{\left(2-0\right)^2+\left(1-0\right)^2}$ $=\sqrt{4+1}$ $=\sqrt{5}$ $\neq 5$

D. $(2, \sqrt{25})$

Using the distance formula, we get:

$\sqrt{\left(2-0\right)^2+\left(\sqrt{25}-0\right)^2}$ $=\sqrt{4+25}$ $=\sqrt{29}$ $\neq 5$

Conclusion

Based on the calculations, we can see that only option A, $(2, \sqrt{21})$ , has a distance from the origin equal to 5 units. Therefore, the point on the circle centered at the origin with a radius of 5 units is:

$(2, \sqrt{21})$

Final Answer

The final answer is:

$A. $(2, \sqrt{21})$<br/> **Q&A: Which Point is on the Circle Centered at the Origin with a Radius of 5 Units?**$

Introduction

In our previous article, we explored the concept of circles and their properties. Specifically, we focused on finding a point that lies on a circle centered at the origin with a radius of 5 units. We used the distance formula to calculate the distance from the origin to each of the given points and determined that the point on the circle is:

$(2, \sqrt{21})$

In this article, we will answer some frequently asked questions related to this topic.

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. It is given by:

$\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}$

Q: How do I use the distance formula to find the distance between two points?

A: To use the distance formula, you need to know the coordinates of the two points. Let's say the coordinates of the two points are $(x_1, y_1)$ and $(x_2, y_2)$ . You can plug these values into the distance formula and calculate the distance.

Q: What is the significance of the radius of a circle?

A: The radius of a circle is the distance from the center of the circle to any point on the circle. In this case, the radius of the circle is 5 units, which means that every point on the circle is 5 units away from the origin.

Q: How do I determine if a point lies on a circle?

A: To determine if a point lies on a circle, you need to calculate the distance from the center of the circle to the point. If the distance is equal to the radius of the circle, then the point lies on the circle.

Q: What if I have a circle with a radius of 10 units? How do I find the point on the circle?

A: To find the point on the circle, you can use the distance formula to calculate the distance from the origin to the point. Since the radius of the circle is 10 units, you can set up an equation using the distance formula and solve for the point.

Q: Can I use the distance formula to find the distance between two points in 3D space?

A: Yes, you can use the distance formula to find the distance between two points in 3D space. The formula is:

$\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2+\left(z_2-z_1\right)^2}$

Q: What if I have a circle with a center at $(3, 4)$ and a radius of 5 units? How do I find the point on the circle?

A: To find the point on the circle, you can use the distance formula to calculate the distance from the center of the circle to the point. Since the radius of the circle is 5 units, you can set up an equation using the distance formula and solve for the point.

Conclusion

In this article, we answered some frequently asked questions related to finding a point on a circle centered at the origin with a radius of 5 units. We also discussed the distance formula and its significance in finding the distance between two points in a coordinate plane.

Final Answer

The final answer is:

A. $(2, \sqrt{21})$