Circle Z is centered at (3, 2) and passes through P(1, 0). How long is the radius o...
Check the final answer first, then review the worked steps.
Problem
Circle Z is centered at (3, 2) and passes through P(1, 0). How long is the radius of circle Z? Simplify any radicals. Does A(5, 4) lie on circle Z?
Answer
\(Radius = 2\sqrt{2}; Yes, point A(5, 4) lies on the circle.\)
Step-by-step solution
- Find the radius of the circle: The radius $r$ of a circle is the distance between its center $(h, k)$ and any point $(x, y)$ on the circle. Given the center $Z(3, 2)$ and point $P(1, 0)$, we use the distance formula: $$r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ Substituting the coordinates: $$r = \sqrt{(1 - 3)^2 + (0 - 2)^2} = \sqrt{(-2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8}$$ Simplifying the radical: $$r = \sqrt{4 \cdot 2} = 2\sqrt{2}$$.
- Determine if point A(5, 4) lies on the circle: A point lies on the circle if its distance from the center is equal to the radius. We calculate the distance $d$ between center $Z(3, 2)$ and point $A(5, 4)$: $$d = \sqrt{(5 - 3)^2 + (4 - 2)^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}$$.
- Conclusion: Since the distance from point $A$ to the center $Z$ is equal to the radius of the circle ($2\sqrt{2} = 2\sqrt{2}$), point $A(5, 4)$ lies on circle $Z$.