Using the Distance Formula

Oct 13, 2025 by leolavanderia.com.br 61 views

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

In mathematics, a circle is a set of points that are all equidistant from a central point known as the center. The distance between the center and any point on the circle is called the radius. In this article, we will explore the concept of a circle centered at the origin with a radius of 5 units and determine which point is on this circle.

Understanding the Circle

A circle centered at the origin with a radius of 5 units can be represented by the equation:

x^2 + y^2 = 25

This equation states that for any point (x, y) on the circle, the sum of the squares of its coordinates is equal to 25.

Using the Distance Formula

To determine which point is on the circle, we can use the distance formula:

Distance = √((x2 - x1)^2 + (y2 - y1)^2)

In this case, we want to find the distance between the origin (0, 0) and each of the given points. We can then compare this distance to the radius of the circle (5 units) to determine which point is on the circle.

Analyzing Option A

Option A is the point (2, √21). To find the distance between the origin and this point, we can use the distance formula:

Distance = √((2 - 0)^2 + (√21 - 0)^2) = √(4 + 21) = √25 = 5

Since the distance between the origin and this point is equal to the radius of the circle (5 units), this point is on the circle.

Analyzing Option B

Option B is the point (2, √23). To find the distance between the origin and this point, we can use the distance formula:

Distance = √((2 - 0)^2 + (√23 - 0)^2) = √(4 + 23) = √27 = √(9 * 3) = 3√3

Since the distance between the origin and this point is not equal to the radius of the circle (5 units), this point is not on the circle.

Analyzing Option C

Option C is the point (2, 1). To find the distance between the origin and this point, we can use the distance formula:

Distance = √((2 - 0)^2 + (1 - 0)^2) = √(4 + 1) = √5

Since the distance between the origin and this point is not equal to the radius of the circle (5 units), this point is not on the circle.

Analyzing Option D

Option D is the point (2, 3). To find the distance between the origin and this point, we can use the distance formula:

Distance = √((2 - 0)^2 + (3 - 0)^2) = √(4 + 9) = √13

Since the distance between the origin and this point is not equal to the radius of the circle (5 units), this point is not on the circle.

In conclusion, the point that is on the circle centered at the origin with a radius of 5 units is Option A, which is the point (2, √21). This point satisfies the equation of the circle and has a distance from the origin equal to the radius of the circle.

The final answer is:

A. (2, √21)