Question

The center of circle C is at (6,2), with a radius of 5. In order to prove that point D (6,-3) lies on the circle, what does the length of segment CD need to be?

Segment CD must have a length of ___

Answer 1

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.

1. The center of the circle C is at the point (6, 2).
2. The coordinates of point D are (6, -3).
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 \( \boxed{5} \).