To determine which point lies on the circle centered at the origin with a radius of 5 units, we use the distance formula [tex]\(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)[/tex] to calculate the distance from the origin [tex]\((0, 0)\)[/tex] to each of the points given.
The points to check are [tex]\((2, \sqrt{21})\)[/tex], [tex]\((2, \sqrt{23})\)[/tex], [tex]\((2, 1)\)[/tex], and [tex]\((2, 3)\)[/tex]:
1. For the point [tex]\((2, \sqrt{21})\)[/tex]:
[tex]\[
\text{Distance} = \sqrt{(2 - 0)^2 + (\sqrt{21} - 0)^2} = \sqrt{2^2 + (\sqrt{21})^2} = \sqrt{4 + 21} = \sqrt{25} = 5
\][/tex]
The distance is 5 units, which matches the radius of the circle.
2. For the point [tex]\((2, \sqrt{23})\)[/tex]:
[tex]\[
\text{Distance} = \sqrt{(2 - 0)^2 + (\sqrt{23} - 0)^2} = \sqrt{2^2 + (\sqrt{23})^2} = \sqrt{4 + 23} = \sqrt{27} \approx 5.196
\][/tex]
The distance is approximately 5.196 units, which is not equal to the radius of the circle.
3. For the point [tex]\((2, 1)\)[/tex]:
[tex]\[
\text{Distance} = \sqrt{(2 - 0)^2 + (1 - 0)^2} = \sqrt{2^2 + 1^2} = \sqrt{4 + 1} = \sqrt{5} \approx 2.236
\][/tex]
The distance is approximately 2.236 units, which is not equal to the radius of the circle.
4. For the point [tex]\((2, 3)\)[/tex]:
[tex]\[
\text{Distance} = \sqrt{(2 - 0)^2 + (3 - 0)^2} = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \approx 3.606
\][/tex]
The distance is approximately 3.606 units, which is not equal to the radius of the circle.
After calculating the distances, we find that the point [tex]\((2, \sqrt{21})\)[/tex] is exactly 5 units away from the origin, matching the circle's radius. Therefore, the point that lies on the circle is:
[tex]\[
(2, \sqrt{21})
\][/tex]
