The equation of the sphere is (x-0.5)^2+(y-0.5)^2+(z-0.5)^2=0.75 and one extremity of its diameter is (1,0,0),then the other extremity of the diameter of the sphere will be_______.A.(0,1,0) B.(0,0,1) C.(0,1,1) D.(1,1,1)

Given the equation of the sphere:

\[
(x - 0.5)^2 + (y - 0.5)^2 + (z - 0.5)^2 = 0.75
\]

The equation of the sphere is in the standard form \((x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\), where \((h, k, l)\) is the center of the sphere, and \(r\) is the radius.

Here, the center of the sphere \((h, k, l)\) is \((0.5, 0.5, 0.5)\) and the radius \(r\) is \(\sqrt{0.75}\).

One extremity of the diameter of the sphere is given as \((1, 0, 0)\).

Let the other extremity of the diameter be \((x, y, z)\). The center of the sphere, which is the midpoint of the diameter, must satisfy the midpoint formula:

\[
\left(\frac{1 + x}{2}, \frac{0 + y}{2}, \frac{0 + z}{2}\right) = (0.5, 0.5, 0.5)
\]

This gives us the following system of equations:

\[
\frac{1 + x}{2} = 0.5
\]

\[
\frac{0 + y}{2} = 0.5
\]

\[
\frac{0 + z}{2} = 0.5
\]

Solving these equations:

1. \( \frac{1 + x}{2} = 0.5 \) gives \( 1 + x = 1 \), so \( x = 0 \).
2. \( \frac{y}{2} = 0.5 \) gives \( y = 1 \).
3. \( \frac{z}{2} = 0.5 \) gives \( z = 1 \).

Therefore, the other extremity of the diameter is \((0, 1, 1)\).

So, the correct answer is **C. \((0, 1, 1)\)**.

Would you like any more details or have any questions? Here are some related questions you might find interesting:

1. How do you find the equation of a sphere given two points on its diameter?
2. What is the general equation of a sphere in 3D space?
3. How do you calculate the distance between two points in 3D space?
4. What is the relationship between the center of a sphere and any two points on its diameter?
5. How can you determine if a point lies inside, on, or outside a sphere?

**Tip:** When solving geometric problems involving spheres, always identify the center and radius first, as they are key to understanding the sphere's properties.

Learn how to determine the other extremity of a sphere's diameter given its equation and one endpoint. This problem involves 3D geometry concepts, including the equation of a sphere and the midpoint formula. Suitable for high school students.

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