Which Point Is On The Circle Centered At The Origin With A Radius Of 5 Units?Distance Formula: $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$A. (2, $\sqrt{21}$) B. (2, $\sqrt{23}$) C. (2, 1) D. (2, 3)

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Which Point is on the Circle Centered at the Origin with a Radius of 5 Units?

In this article, we will explore the concept of circles and their properties. Specifically, we will focus on finding a point 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 is a mathematical concept used to find the distance between two points in a coordinate plane. It is given by the equation:

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

This formula calculates the distance between two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) in a coordinate plane.

A circle is a set of points that are equidistant from a central point called the center. The distance from the center to any point on the circle is called the radius. In this case, the circle is centered at the origin, which means that the center is at the point (0,0)(0, 0). The radius of the circle is given as 5 units.

To find the point on the circle, we need to use the distance formula. We know that the center of the circle is at the origin, so the coordinates of the center are (0,0)(0, 0). We also know that the radius of the circle is 5 units. Let's assume that the point on the circle has coordinates (x,y)(x, y).

Using the distance formula, we can set up the equation:

(x−0)2+(y−0)2=5\sqrt{(x-0)^2+(y-0)^2} = 5

Simplifying the equation, we get:

x2+y2=5\sqrt{x^2+y^2} = 5

Squaring both sides of the equation, we get:

x2+y2=25x^2+y^2 = 25

Now, let's look at the answer choices and see which one satisfies the equation.

Answer Choice A

The first answer choice is (2,21)(2, \sqrt{21}). Let's substitute these values into the equation:

x2+y2=22+(21)2x^2+y^2 = 2^2+(\sqrt{21})^2

Simplifying the equation, we get:

x2+y2=4+21x^2+y^2 = 4+21

x2+y2=25x^2+y^2 = 25

This answer choice satisfies the equation, so it is a possible solution.

Answer Choice B

The second answer choice is (2,23)(2, \sqrt{23}). Let's substitute these values into the equation:

x2+y2=22+(23)2x^2+y^2 = 2^2+(\sqrt{23})^2

Simplifying the equation, we get:

x2+y2=4+23x^2+y^2 = 4+23

x2+y2=27x^2+y^2 = 27

This answer choice does not satisfy the equation, so it is not a possible solution.

Answer Choice C

The third answer choice is (2,1)(2, 1). Let's substitute these values into the equation:

x2+y2=22+12x^2+y^2 = 2^2+1^2

Simplifying the equation, we get:

x2+y2=4+1x^2+y^2 = 4+1

x2+y2=5x^2+y^2 = 5

This answer choice does not satisfy the equation, so it is not a possible solution.

Answer Choice D

The fourth answer choice is (2,3)(2, 3). Let's substitute these values into the equation:

x2+y2=22+32x^2+y^2 = 2^2+3^2

Simplifying the equation, we get:

x2+y2=4+9x^2+y^2 = 4+9

x2+y2=13x^2+y^2 = 13

This answer choice does not satisfy the equation, so it is not a possible solution.

In this article, we used the distance formula to find a point on a circle centered at the origin with a radius of 5 units. We looked at four answer choices and found that only one of them satisfied the equation. The correct answer is:

  • A. (2, 21\sqrt{21})

This answer choice satisfies the equation, so it is the correct solution.

In this article, we explored the concept of circles and their properties. We used the distance formula to find a point on a circle centered at the origin with a radius of 5 units. We looked at four answer choices and found that only one of them satisfied the equation. This article demonstrates the importance of using mathematical concepts to solve problems and find solutions.
Q&A: Understanding Circles and the Distance Formula

In our previous article, we explored the concept of circles and their properties. We used the distance formula to find a point on a circle centered at the origin with a radius of 5 units. In this article, we will answer some frequently asked questions about circles and the distance formula.

Q: What is the distance formula?

A: The distance formula is a mathematical concept used to find the distance between two points in a coordinate plane. It is given by the equation:

(x2−x1)2+(y2−y1)2\sqrt{(x_2-x_1)^2+(y_2-y_1)^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 (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2). You can plug these values into the distance formula to find the distance between the two points.

Q: What is the center of a circle?

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

Q: What is 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.

Q: How do I find the point on a circle using the distance formula?

A: To find the point on a circle using the distance formula, you need to know the coordinates of the center of the circle and the radius of the circle. Let's say the center of the circle is at the point (0,0)(0, 0) and the radius of the circle is 5 units. You can use the distance formula to find the point on the circle.

Q: What is the equation of a circle?

A: The equation of a circle is given by the equation:

(x−h)2+(y−k)2=r2(x-h)^2+(y-k)^2=r^2

where (h,k)(h, k) is the center of the circle and rr is the radius of the circle.

Q: How do I find the equation of a circle?

A: To find the equation of a circle, you need to know the coordinates of the center of the circle and the radius of the circle. Let's say the center of the circle is at the point (h,k)(h, k) and the radius of the circle is rr. You can plug these values into the equation of a circle to find the equation of the circle.

Q: What is the relationship between the distance formula and the equation of a circle?

A: The distance formula and the equation of a circle are related. The distance formula can be used to find the distance between two points on a circle, and the equation of a circle can be used to find the equation of a circle.

Q: How do I use the distance formula and the equation of a circle to solve problems?

A: To use the distance formula and the equation of a circle to solve problems, you need to know the coordinates of the points and the radius of the circle. You can plug these values into the distance formula and the equation of a circle to find the solution to the problem.

In this article, we answered some frequently asked questions about circles and the distance formula. We explored the concept of circles and their properties, and we used the distance formula to find a point on a circle centered at the origin with a radius of 5 units. We also discussed the relationship between the distance formula and the equation of a circle. This article demonstrates the importance of using mathematical concepts to solve problems and find solutions.

In this article, we explored the concept of circles and their properties. We used the distance formula to find a point on a circle centered at the origin with a radius of 5 units. We also answered some frequently asked questions about circles and the distance formula. This article demonstrates the importance of using mathematical concepts to solve problems and find solutions.