Recall The Definition Of A Circle As The Set Of All Points Equidistant From A Center Point. Given The Center \[$(3, 5)\$\], Prove That \[$(0, 5)\$\] And \[$(3, 2)\$\] Are Points On The Circle And Describe The Radius Of The

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A circle is a fundamental concept in geometry, defined as the set of all points equidistant from a center point. This definition forms the basis of our understanding of circles and their properties. In this article, we will explore the concept of a circle, its definition, and how to prove that certain points lie on its surface. We will also calculate the radius of the circle given its center point.

The Definition of a Circle

A circle is a set of points that are all equidistant from a central point, known as the center. This means that every point on the circle is at the same distance from the center point. The distance between the center point and any point on the circle is called the radius of the circle.

The Center of the Circle

The center of the circle is given as {(3, 5)$}$. This is the point from which all other points on the circle are equidistant.

Proving Points on the Circle

To prove that a point lies on the circle, we need to show that it is equidistant from the center point. Let's consider the two points {(0, 5)$}$ and {(3, 2)$}$.

Point {(0, 5)$]

To prove that [(0, 5)\$} lies on the circle, we need to calculate the distance between this point and the center point {(3, 5)$}$. We can use the distance formula to calculate this distance:

d=(x2−x1)2+(y2−y1)2{d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}}

where (x1,y1){(x_1, y_1)} is the center point {(3, 5)$}$ and (x2,y2){(x_2, y_2)} is the point {(0, 5)$}$.

Plugging in the values, we get:

d=(0−3)2+(5−5)2{d = \sqrt{(0 - 3)^2 + (5 - 5)^2}} d=(−3)2+02{d = \sqrt{(-3)^2 + 0^2}} d=9+0{d = \sqrt{9 + 0}} d=9{d = \sqrt{9}} d=3{d = 3}

This means that the distance between the center point {(3, 5)$}$ and the point {(0, 5)$}$ is 3 units. Since this distance is equal to the radius of the circle, we can conclude that {(0, 5)$}$ lies on the circle.

Point {(3, 2)$]

To prove that [(3, 2)\$} lies on the circle, we need to calculate the distance between this point and the center point {(3, 5)$}$. We can use the distance formula to calculate this distance:

d=(x2−x1)2+(y2−y1)2{d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}}

where (x1,y1){(x_1, y_1)} is the center point {(3, 5)$}$ and (x2,y2){(x_2, y_2)} is the point {(3, 2)$}$.

Plugging in the values, we get:

d=(3−3)2+(2−5)2{d = \sqrt{(3 - 3)^2 + (2 - 5)^2}} d=02+(−3)2{d = \sqrt{0^2 + (-3)^2}} d=0+9{d = \sqrt{0 + 9}} d=9{d = \sqrt{9}} d=3{d = 3}

This means that the distance between the center point {(3, 5)$}$ and the point {(3, 2)$}$ is 3 units. Since this distance is equal to the radius of the circle, we can conclude that {(3, 2)$}$ lies on the circle.

The Radius of the Circle

The radius of the circle is the distance between the center point and any point on the circle. We have already calculated this distance for the two points {(0, 5)$}$ and {(3, 2)$}$, and we found that it is equal to 3 units.

Therefore, the radius of the circle is 3 units.

Conclusion

In this article, we have explored the concept of a circle, its definition, and how to prove that certain points lie on its surface. We have also calculated the radius of the circle given its center point. We have shown that the points {(0, 5)$}$ and {(3, 2)$}$ lie on the circle and that the radius of the circle is 3 units.

References

In our previous article, we explored the concept of a circle, its definition, and how to prove that certain points lie on its surface. We also calculated the radius of the circle given its center point. In this article, we will answer some frequently asked questions about circles to help you better understand this fundamental concept in geometry.

Q: What is the definition of a circle?

A: A circle is a set of points that are all equidistant from a central point, known as the center. This means that every point on the circle is at the same distance from the center point.

Q: What is the center of a circle?

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

Q: What is the radius of a circle?

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

Q: How do you calculate the radius of a circle?

A: To calculate the radius of a circle, you need to know the distance between the center point and any point on the circle. You can use the distance formula to calculate this distance:

d=(x2−x1)2+(y2−y1)2{d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}}

where (x1,y1){(x_1, y_1)} is the center point and (x2,y2){(x_2, y_2)} is the point on the circle.

Q: What is the circumference of a circle?

A: The circumference of a circle is the distance around the circle. It is a measure of the total distance around the circle.

Q: How do you calculate the circumference of a circle?

A: To calculate the circumference of a circle, you need to know the radius of the circle. You can use the formula:

C=2Ï€r{C = 2\pi r}

where C{C} is the circumference and r{r} is the radius.

Q: What is the area of a circle?

A: The area of a circle is the amount of space inside the circle. It is a measure of the size of the circle.

Q: How do you calculate the area of a circle?

A: To calculate the area of a circle, you need to know the radius of the circle. You can use the formula:

A=Ï€r2{A = \pi r^2}

where A{A} is the area and r{r} is the radius.

Q: What is the difference between a circle and an ellipse?

A: A circle is a set of points that are all equidistant from a central point, while an ellipse is a set of points that are all equidistant from two central points.

Q: Can a circle have a negative radius?

A: No, a circle cannot have a negative radius. The radius of a circle is always a positive value.

Q: Can a circle have a zero radius?

A: No, a circle cannot have a zero radius. A circle with a zero radius would be a point, not a circle.

Conclusion

In this article, we have answered some frequently asked questions about circles to help you better understand this fundamental concept in geometry. We have covered topics such as the definition of a circle, the center and radius of a circle, and how to calculate the circumference and area of a circle. We hope that this article has been helpful in your understanding of circles.

References