Comparing the Distance to the Radius

Oct 13, 2025 by magicalceylontours.com 48 views

Introduction

In geometry, a circle is defined as the set of all points in a plane that are equidistant from a central point, known as the center. The distance from the center to any point on the circle is called the radius. In this article, we will explore whether a given point lies on a circle centered at a specific point.

The Circle and Its Center

The circle in question is centered at the point ${$ (0, -9)$}$. This means that every point on the circle is equidistant from the point ${$ (0, -9)$}$. To determine whether a point lies on the circle, we need to calculate the distance from the center to the point and compare it to the radius of the circle.

The Point in Question

The point in question is ${$ (8, \sqrt{17})$}$. To determine whether this point lies on the circle, we need to calculate the distance from the center to this point.

Calculating the Distance

The distance between two points ${$ (x_1, y_1)$}$ and ${$ (x_2, y_2)$}$ in a plane is given by the formula:

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

In this case, the center of the circle is ${$ (0, -9)$}$ and the point in question is ${$ (8, \sqrt{17})$}$. Plugging these values into the formula, we get:

d = \sqrt{(8 - 0)^2 + (\sqrt{17} - (-9))^2}

Simplifying the expression, we get:

d = \sqrt{64 + (\sqrt{17} + 9)^2}

Expanding the squared term, we get:

d = \sqrt{64 + 17 + 18\sqrt{17} + 81}

Combining like terms, we get:

d = \sqrt{146 + 18\sqrt{17}}

Comparing the Distance to the Radius

To determine whether the point lies on the circle, we need to compare the distance from the center to the point to the radius of the circle. However, we are not given the radius of the circle. Therefore, we cannot determine whether the point lies on the circle based on the distance alone.

Conclusion

In conclusion, we cannot determine whether the point ${$ (8, \sqrt{17})$}$ lies on the circle centered at ${$ (0, -9)$}$ based on the distance from the center to the point. We need more information, such as the radius of the circle, to make a determination.

The Final Answer

The final answer is: No, we cannot determine whether the point lies on the circle based on the distance alone.

Additional Information

If you are interested in learning more about circles and their properties, I recommend checking out the following resources:

Introduction

In our previous article, we explored whether a given point lies on a circle centered at a specific point. In this article, we will answer some frequently asked questions about circles and points.

Q: What is the definition of a circle?

A: A circle is a set of all points in a plane that are equidistant from a central point, known as the center. The distance from the center to any point on the circle is called the radius.

Q: How do I calculate the distance from the center to a point on a circle?

A: To calculate the distance from the center to a point on a circle, you can use the formula:

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

where ${$ (x_1, y_1)$}$ is the center of the circle and ${$ (x_2, y_2)$}$ is the point in question.

Q: What is the radius of a circle?

A: The radius of a circle is the distance from the center to any point on the circle. It is a measure of the size of the circle.

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

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

Q: What if I don't know the radius of the circle?

A: If you don't know the radius of the circle, then you cannot determine whether a point lies on the circle based on the distance alone. You need more information, such as the radius of the circle, to make a determination.

Q: Can a point lie on a circle if it is not equidistant from the center?

A: No, a point can only lie on a circle if it is equidistant from the center. If a point is not equidistant from the center, then it does not lie on the circle.

Q: What is the relationship between the center and the points on a circle?

A: The center of a circle is the point that is equidistant from all points on the circle. The points on a circle are all equidistant from the center.

Q: Can a circle have more than one center?

A: No, a circle can only have one center. The center of a circle is a unique point that is equidistant from all points on the circle.

Q: What is the relationship between the radius and the points on a circle?

A: The radius of a circle is the distance from the center to any point on the circle. The points on a circle are all equidistant from the center, and the distance from the center to any point on the circle is equal to the radius.

Conclusion

In conclusion, we have answered some frequently asked questions about circles and points. We hope that this article has been helpful in understanding the properties of circles and points.

Additional Information

If you are interested in learning more about circles and their properties, I recommend checking out the following resources:

I hope this article has been helpful in answering your questions about circles and points. If you have any further questions or comments, please feel free to leave them below.