To provide a detailed, step-by-step solution for this proof, we need to establish that the point [tex]\((1, \sqrt{3})\)[/tex] lies on the circle centered at the origin and containing the point [tex]\((0,2)\)[/tex]. Here is the justification for each statement leading up to the conclusion:
1. Statement: A circle is centered at [tex]\((0,0)\)[/tex] and contains the point [tex]\((0,2)\)[/tex].
- Justification: Given.
2. Statement: The radius of the circle is the distance from [tex]\((0,0)\)[/tex] to [tex]\((0,2)\)[/tex].
- Justification: Definition of radius.
3. Statement: The distance from [tex]\((0,0)\)[/tex] to [tex]\((0,2)\)[/tex] is [tex]\(\sqrt{(0-0)^2 + (2-0)^2} = \sqrt{2^2} = 2\)[/tex].
- Justification: Distance formula.
Having established the radius of the circle, we move on to verify whether the point [tex]\((1, \sqrt{3})\)[/tex] lies on the same circle. We do this by calculating its distance to the center [tex]\((0,0)\)[/tex] and checking if it equals the radius of the circle.
4. Statement: If [tex]\((1, \sqrt{3})\)[/tex] lies on the circle, it must be the same distance from the center [tex]\((0,0)\)[/tex] as [tex]\((0,2)\)[/tex].
- Justification: Points on a circle are equidistant from the center by definition.
Next, we calculate the distance from [tex]\((0,0)\)[/tex] to [tex]\((1, \sqrt{3})\)[/tex].
5. Statement: The distance from [tex]\((0,0)\)[/tex] to [tex]\((1, \sqrt{3})\)[/tex] is [tex]\(\sqrt{(0-1)^2 + (0-\sqrt{3})^2} = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1+3} = 2\)[/tex].
- Justification: Distance formula.
Finally, we draw our conclusion based on the computed distances.
6. Statement: Since [tex]\((1, \sqrt{3})\)[/tex] is 2 units from [tex]\((0,0)\)[/tex], it lies on a circle that is centered at the origin and contains the point [tex]\((0,2)\)[/tex].
- Justification: Definition of a circle.
Thus, we have shown step-by-step that the point [tex]\((1, \sqrt{3})\)[/tex] is indeed on the circle centered at [tex]\((0,0)\)[/tex] with radius 2.
