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 this article, we will explore the concept of a circle centered at the origin with a radius of 5 units. We will use the distance formula to determine which point is on the circle. The distance formula is a fundamental concept in mathematics that is used to calculate the distance between two points in a coordinate plane.

Understanding the Distance Formula

The distance formula is given by:

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

This formula calculates the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ in a coordinate plane. To use this formula, we need to substitute the coordinates of the two points into the formula and calculate the result.

The Circle Centered at the Origin with a Radius of 5 Units

The circle centered at the origin with a radius of 5 units has the equation:

$x^2 + y^2 = 25$

This equation represents a circle with a center at the origin (0, 0) and a radius of 5 units. Any point that satisfies this equation is on the circle.

Using the Distance Formula to Find the Point on the Circle

We are given four options for points that may be on the circle:

A. (2, $\sqrt{21}$) B. (2, $\sqrt{23}$) C. (2, 1) D. (2, 3)

To determine which point is on the circle, we need to use the distance formula to calculate the distance between each point and the origin. If the distance is equal to the radius of the circle (5 units), then the point is on the circle.

Calculating the Distance for Each Point

Let's calculate the distance for each point using the distance formula:

Point A: (2, $\sqrt{21}$)

The distance between the point (2, $\sqrt{21}$) and the origin (0, 0) is:

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

Since the distance is equal to the radius of the circle (5 units), point A is on the circle.

Point B: (2, $\sqrt{23}$)

The distance between the point (2, $\sqrt{23}$) and the origin (0, 0) is:

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

Since the distance is greater than the radius of the circle (5 units), point B is not on the circle.

Point C: (2, 1)

The distance between the point (2, 1) and the origin (0, 0) is:

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

Since the distance is less than the radius of the circle (5 units), point C is not on the circle.

Point D: (2, 3)

The distance between the point (2, 3) and the origin (0, 0) is:

$\sqrt{(2-0)^2+(3-0)^2}$ $=\sqrt{4+9}$ $=\sqrt{13}$ $<5$

Since the distance is less than the radius of the circle (5 units), point D is not on the circle.

Conclusion

In conclusion, the point that is on the circle centered at the origin with a radius of 5 units is point A: (2, $\sqrt{21}$). The distance formula was used to calculate the distance between each point and the origin, and only point A had a distance equal to the radius of the circle (5 units).

Final Answer

The final answer is:

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Introduction

In our previous article, we explored the concept of a circle centered at the origin with a radius of 5 units. We used the distance formula to determine 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:

$x^2 + y^2 = 25$

This equation represents a circle with a center at the origin (0, 0) and a radius of 5 units.

Q: How do I use the distance formula to find the point on the circle?

A: To use the distance formula to find the point on the circle, you need to substitute the coordinates of the point into the formula and calculate the result. The distance formula is given by:

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

In this case, the coordinates of the point are (x, y) and the coordinates of the origin are (0, 0). If the distance is equal to the radius of the circle (5 units), then the point is on the circle.

Q: What is the distance formula?

A: The distance formula is a fundamental concept in mathematics that is used to calculate the distance between two points in a coordinate plane. The distance formula is given by:

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

This formula calculates the distance between two points (x1, y1) and (x2, y2) in a coordinate plane.

Q: How do I calculate the distance between two points using the distance formula?

A: To calculate the distance between two points using the distance formula, you need to substitute the coordinates of the two points into the formula and calculate the result. For example, if you want to calculate the distance between the points (2, 3) and (4, 5), you would substitute the coordinates into the formula as follows:

$\sqrt{(4-2)^2+(5-3)^2}$ $=\sqrt{2^2+2^2}$ $=\sqrt{4+4}$ $=\sqrt{8}$ $=2\sqrt{2}$

Q: What is the significance of the distance formula in mathematics?

A: The distance formula is a fundamental concept in mathematics that is used to calculate the distance between two points in a coordinate plane. It is used in a wide range of applications, including geometry, trigonometry, and calculus.

Q: Can I use the distance formula to calculate the distance between two points in three-dimensional space?

A: Yes, you can use the distance formula to calculate the distance between two points in three-dimensional space. The distance formula is given by:

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

This formula calculates the distance between two points (x1, y1, z1) and (x2, y2, z2) in three-dimensional space.

Conclusion

In conclusion, the distance formula is a fundamental concept in mathematics that is used to calculate the distance between two points in a coordinate plane. It is used in a wide range of applications, including geometry, trigonometry, and calculus. We hope that this article has helped to answer some of your questions related to the distance formula and the circle centered at the origin with a radius of 5 units.

Final Answer

The final answer is:

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