To determine if point D(6, -3) lies on circle C, we need to check if the distance from the center of the circle C(6, 2) to point D is equal to the radius of the circle, which is 5.
The length of segment \( CD \) can be calculated using the distance formula:
\[ CD = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
In this case:
- \( C(6, 2) \)
- \( D(6, -3) \)
Substituting the values into the formula:
\[ CD = \sqrt{(6 - 6)^2 + (-3 - 2)^2} \] \[ = \sqrt{0^2 + (-5)^2} \] \[ = \sqrt{0 + 25} \] \[ = \sqrt{25} \] \[ = 5 \]
The length of segment \( CD \) must be 5. Thus, point D lies on circle C.
Final Answer: Segment \( CD \) must have a length of 5.