A rectangular parallelepiped is formed by planes drawn through the points `(2,3,5) and (5,9,7)` parallel to the coordinate planes. The length of a diagonal of the parallelepiped is

Options

A. 7 units
B. `sqrt(38)` units
C. `sqrt( 155)` units
D. none of these

Answer

To find the length of the diagonal of the rectangular parallelepiped formed by the points (2,3,5) and (5,9,7), we can follow these steps:

### Step 1: Identify the coordinates of the points
The two points given are:
- Point A: (2, 3, 5)
- Point B: (5, 9, 7)

### Step 2: Use the distance formula
The length of the diagonal of the rectangular parallelepiped can be calculated using the distance formula in three-dimensional space. The distance \(d\) between two points \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) is given by:

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

### Step 3: Substitute the coordinates into the formula
Substituting the coordinates of points A and B into the distance formula:

\[ d = \sqrt{(5 - 2)^2 + (9 - 3)^2 + (7 - 5)^2} \]

### Step 4: Calculate the differences
Now, calculate the differences:

- \(5 - 2 = 3\)
- \(9 - 3 = 6\)
- \(7 - 5 = 2\)

### Step 5: Square the differences
Next, square each of these differences:

- \(3^2 = 9\)
- \(6^2 = 36\)
- \(2^2 = 4\)

### Step 6: Add the squared differences
Now, add the squared differences:

\[ 9 + 36 + 4 = 49 \]

### Step 7: Take the square root
Finally, take the square root of the sum to find the length of the diagonal:

\[ d = \sqrt{49} = 7 \]

### Conclusion
The length of the diagonal of the rectangular parallelepiped is **7 units**.

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