To determine which point lies on the circle centered at the origin with a radius of 5 units, we need to calculate the distance from the origin [tex]\((0,0)\)[/tex] to each of the given points and see which one matches the radius of the circle.
Given circle radius: [tex]\( r = 5 \)[/tex] units.
The distance formula is:
[tex]\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Since the circle is centered at the origin [tex]\((0, 0)\)[/tex], we use the formula [tex]\( \text{Distance} = \sqrt{x^2 + y^2} \)[/tex].
Let's calculate the distance for each point:
### 1. Point [tex]\((2, \sqrt{21})\)[/tex]:
[tex]\[
\text{Distance} = \sqrt{2^2 + (\sqrt{21})^2} = \sqrt{4 + 21} = \sqrt{25} = 5
\][/tex]
### 2. Point [tex]\((2, \sqrt{23})\)[/tex]:
[tex]\[
\text{Distance} = \sqrt{2^2 + (\sqrt{23})^2} = \sqrt{4 + 23} = \sqrt{27} \approx 5.196
\][/tex]
### 3. Point [tex]\((2, 1)\)[/tex]:
[tex]\[
\text{Distance} = \sqrt{2^2 + 1^2} = \sqrt{4 + 1} = \sqrt{5} \approx 2.236
\][/tex]
### 4. Point [tex]\((2, 3)\)[/tex]:
[tex]\[
\text{Distance} = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \approx 3.606
\][/tex]
Now we compare each distance with the given radius (5 units):
1. The distance for point [tex]\((2, \sqrt{21})\)[/tex] is [tex]\(5\)[/tex], which matches the radius.
2. The distance for point [tex]\((2, \sqrt{23})\)[/tex] is approximately [tex]\(5.196\)[/tex], which does not match the radius.
3. The distance for point [tex]\((2, 1)\)[/tex] is approximately [tex]\(2.236\)[/tex], which does not match the radius.
4. The distance for point [tex]\((2, 3)\)[/tex] is approximately [tex]\(3.606\)[/tex], which does not match the radius.
Therefore, the point [tex]\((2, \sqrt{21})\)[/tex] lies on the circle centered at the origin with a radius of 5 units.
