To determine if point D(6, -3) lies on circle C, we need to check the distance between the center of the circle (point C) at (6, 2) and point D. The length of segment CD must be equal to the radius of the circle, which is 5.
Firstly, we will calculate the distance \(CD\) using the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Where:
- \((x_1, y_1)\) are the coordinates of point C (center of the circle),
- \((x_2, y_2)\) are the coordinates of point D.
Substituting the coordinates:
- \(C(6, 2)\)
- \(D(6, -3)\)
Thus, \[ d = \sqrt{(6 - 6)^2 + (-3 - 2)^2} \] \[ d = \sqrt{0^2 + (-5)^2} \] \[ d = \sqrt{0 + 25} \] \[ d = \sqrt{25} \] \[ d = 5 \]
Since the calculated length of segment \(CD\) is indeed 5, which is equal to the radius of the circle, it confirms that point D lies on the circle.
Therefore, the length of segment \(CD\) must be \(5\).
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