To determine whether point \( D(6, -3) \) lies on circle \( C \) with center \( (6, 2) \) and radius \( 5 \), we need to find the distance \( CD \) between the center of the circle \( C(6, 2) \) and point \( D(6, -3) \).
We can use the distance formula, which is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
In this case:
- \( (x_1, y_1) = (6, 2) \) (the center of the circle)
- \( (x_2, y_2) = (6, -3) \) (point D)
Substituting the values into the distance formula:
\[ d = \sqrt{(6 - 6)^2 + (-3 - 2)^2} \]
This simplifies to:
\[ d = \sqrt{(0)^2 + (-5)^2} \] \[ d = \sqrt{0 + 25} \] \[ d = \sqrt{25} \] \[ d = 5 \]
Since the radius of the circle is \( 5 \), and the length of segment \( CD \) is also \( 5 \), point \( D \) indeed lies on the circle.
Therefore, the length of segment \( CD \) must be \( 5 \).
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