To determine which point is on the circle centered at the origin (0, 0) with a radius of 5 units, we need to calculate the distance of each point from the origin using the distance formula and then compare the result with the circle's radius.
The distance formula between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Here, our circle is centered at the origin [tex]\((0, 0)\)[/tex] and has a radius of 5 units, so we will calculate the distance of each point from the origin.
1. Point [tex]\((2, \sqrt{21})\)[/tex]:
[tex]\[
\sqrt{(2 - 0)^2 + (\sqrt{21} - 0)^2} = \sqrt{2^2 + (\sqrt{21})^2} = \sqrt{4 + 21} = \sqrt{25} = 5
\][/tex]
The distance is 5, so this point lies on the circle.
2. Point [tex]\((2, \sqrt{23})\)[/tex]:
[tex]\[
\sqrt{(2 - 0)^2 + (\sqrt{23} - 0)^2} = \sqrt{2^2 + (\sqrt{23})^2} = \sqrt{4 + 23} = \sqrt{27} \approx 5.196
\][/tex]
The distance is approximately 5.196, which is greater than 5, so this point does not lie on the circle.
3. Point [tex]\((2, 1)\)[/tex]:
[tex]\[
\sqrt{(2 - 0)^2 + (1 - 0)^2} = \sqrt{2^2 + 1^2} = \sqrt{4 + 1} = \sqrt{5} \approx 2.236
\][/tex]
The distance is approximately 2.236, which is less than 5, so this point does not lie on the circle.
4. Point [tex]\((2, 3)\)[/tex]:
[tex]\[
\sqrt{(2 - 0)^2 + (3 - 0)^2} = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \approx 3.606
\][/tex]
The distance is approximately 3.606, which is less than 5, so this point does not lie on the circle.
Based on the calculations, the point that lies on the circle centered at the origin with a radius of 5 units is:
[tex]\[
(2, \sqrt{21})
\][/tex]
