To determine if point D (6, -3) lies on circle C with center at (6, 2) and a radius of 5, we need to calculate the length of segment CD and compare it to the radius of the circle.
- The center of the circle C is at the point (6, 2).
- The coordinates of point D are (6, -3).
- To find the length of segment CD, we use the distance formula, which is given by:
\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Where (x₁, y₁) is the center of the circle and (x₂, y₂) is the point D:
- Here, (x₁, y₁) = (6, 2)
- And (x₂, y₂) = (6, -3)
Substituting the values into the distance formula:
\[ \text{Distance} = \sqrt{(6 - 6)^2 + (-3 - 2)^2} \]
This simplifies to:
\[ \text{Distance} = \sqrt{0 + (-5)^2} = \sqrt{25} = 5 \]
Since the radius of the circle is also 5, the length of segment CD must be equal to the radius of the circle for point D to lie on the circle.
Thus, the length of segment CD must be \( \boxed{5} \).
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