Find the center and radius of a sphere

Question

Find the center and radius of a sphere that has $(1,-2,4)$ and $(3,4,-12)$ as endpoints of a diameter.

Okay. I Understand how to find radius. It is merely the distance formula applied to a sphere. However, I am having trouble trying to find the center point. Help?

Answer 1

HINT:

The center is the middle point of any diameter.

Answer 2

Use the midpoint formula. The center is (1, 1, -4).

Answer 3

A diameter of a sphere is a segment that starts at the surface of the sphere, goes straight through the center of the sphere, and ends at the surface of the sphere opposite where it started.

So the center of the sphere is on the line between the two given points.

Knowing that the surface of the sphere is the same distance from the center in any direction, the center is just halfway between the two ends of the diameter. That point is easy to find.

Answer 4

Remember that the center of a circle (or sphere in your case) is the midpoint of its diameter:

$$x_{mid}= \frac{x_1+x_2}{2}$$ $$y_{mid}= \frac{y_1+y_2}{2}$$ $$z_{mid}= \frac{z_1+z_2}{2}$$