To determine which given point lies on the circle centered at the origin [tex]\((0,0)\)[/tex] with a radius of 5 units, we use the distance formula:
[tex]\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Here, the center of the circle is [tex]\((0,0)\)[/tex], and the radius is 5 units. The equation for the distance from any point [tex]\((x,y)\)[/tex] to the center can be set equal to the radius:
[tex]\[
\sqrt{(x - 0)^2 + (y - 0)^2} = 5
\][/tex]
This simplifies to:
[tex]\[
\sqrt{x^2 + y^2} = 5
\][/tex]
By squaring both sides, we get:
[tex]\[
x^2 + y^2 = 25
\][/tex]
Now, we will test each of the given points to see if it satisfies this equation.
### First Point: [tex]\((2, \sqrt{21})\)[/tex]
[tex]\[
x = 2, \quad y = \sqrt{21}
\][/tex]
[tex]\[
x^2 + y^2 = 2^2 + (\sqrt{21})^2 = 4 + 21 = 25
\][/tex]
This point satisfies the equation.
### Second Point: [tex]\((2, \sqrt{23})\)[/tex]
[tex]\[
x = 2, \quad y = \sqrt{23}
\][/tex]
[tex]\[
x^2 + y^2 = 2^2 + (\sqrt{23})^2 = 4 + 23 = 27
\][/tex]
This point does not satisfy the equation.
### Third Point: [tex]\((2, 1)\)[/tex]
[tex]\[
x = 2, \quad y = 1
\][/tex]
[tex]\[
x^2 + y^2 = 2^2 + 1^2 = 4 + 1 = 5
\][/tex]
This point does not satisfy the equation.
### Fourth Point: [tex]\((2, 3)\)[/tex]
[tex]\[
x = 2, \quad y = 3
\][/tex]
[tex]\[
x^2 + y^2 = 2^2 + 3^2 = 4 + 9 = 13
\][/tex]
This point does not satisfy the equation.
The only point that satisfies the equation [tex]\(x^2 + y^2 = 25\)[/tex] is [tex]\((2, \sqrt{21})\)[/tex].
Therefore, the point that lies on the circle centered at the origin with a radius of 5 units is:
[tex]\[
(2, \sqrt{21})
\][/tex]
