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. `2 sqrt(3) `
B. `3 sqrt(2)`
C. `sqrt(2)`
D. `sqrt(3)`

Answer

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
The two points given are:
- Point P: (5, 8, 10)
- Point Q: (3, 6, 8)

### Step 2: Use the Distance Formula
The length of the diagonal (distance between points P and Q) can be calculated using the distance formula in three dimensions:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
where \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) are the coordinates of points P and Q respectively.

### Step 3: Substitute the Coordinates
Substituting the coordinates of points P and Q into the distance formula:
\[ d = \sqrt{(3 - 5)^2 + (6 - 8)^2 + (8 - 10)^2} \]

### Step 4: Calculate Each Component
Now, calculate each component:
- \(3 - 5 = -2\) so \((-2)^2 = 4\)
- \(6 - 8 = -2\) so \((-2)^2 = 4\)
- \(8 - 10 = -2\) so \((-2)^2 = 4\)

### Step 5: Sum the Squares
Now, sum these squares:
\[ d = \sqrt{4 + 4 + 4} = \sqrt{12} \]

### Step 6: Simplify the Square Root
Simplifying \(\sqrt{12}\):
\[ \sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3} \]

### Final Answer
Thus, the length of the diagonal of the parallelepiped is:
\[ \boxed{2\sqrt{3}} \]

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