If a parallelopiped is formed by planes drawn through the points ( 5,8,10) and (3,6,8) parallel to the coordinate planes, then the length of diagonal of the parallelopiped is

Options

A. `2sqrt3`
B. `3sqrt2`
C. `sqrt2`
D. `sqrt3`

Explanation

To find the length of the diagonal of the parallelepiped formed by the points (5, 8, 10) and (3, 6, 8), we can follow these steps:

### Step 1: Identify the Points
We have two points:
- Point A: \( A(5, 8, 10) \)
- Point B: \( B(3, 6, 8) \)

### Step 2: Use the Distance Formula
The length of the diagonal of the parallelepiped can be calculated using the distance formula in three-dimensional space, which is given by:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]

### Step 3: Assign Coordinates
Here, we assign:
- \( (x_1, y_1, z_1) = (5, 8, 10) \)
- \( (x_2, y_2, z_2) = (3, 6, 8) \)

### Step 4: Substitute the Coordinates into the Formula
Now, we substitute the coordinates into the distance formula:

\[ d = \sqrt{(3 - 5)^2 + (6 - 8)^2 + (8 - 10)^2} \]

### Step 5: Calculate Each Component
Calculate each term inside the square root:

1. \( (3 - 5)^2 = (-2)^2 = 4 \)
2. \( (6 - 8)^2 = (-2)^2 = 4 \)
3. \( (8 - 10)^2 = (-2)^2 = 4 \)

### Step 6: Sum the Components
Now sum these values:

\[ d = \sqrt{4 + 4 + 4} = \sqrt{12} \]

### Step 7: Simplify the Result
We can simplify \( \sqrt{12} \):

\[ \sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3} \]

### Final Answer
Thus, the length of the diagonal of the parallelepiped is:

\[ \boxed{2\sqrt{3}} \]

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