A rectangular parallelopiped is formed by drawing planes through the points `(1,2,5) and (-1,-1,-1)` parallel to the coordinate planes. Find the length of the diagnol of the parallelopiped.

Answer

To find the length of the diagonal of a rectangular parallelepiped formed by drawing planes through the points \( P(1, 2, 5) \) and \( Q(-1, -1, -1) \) parallel to the coordinate planes, we can follow these steps:

### Step 1: Identify the coordinates of the points
We have two points:
- \( P(1, 2, 5) \)
- \( Q(-1, -1, -1) \)

### Step 2: Calculate the lengths of the edges of the parallelepiped
The edges of the parallelepiped can be calculated as follows:

1. **Length along the x-axis (PA)**:
\[ PA = |x_1 - x_2| = |1 - (-1)| = |1 + 1| = |2| = 2 \]

2. **Length along the y-axis (PC)**:
\[ PC = |y_1 - y_2| = |2 - (-1)| = |2 + 1| = |3| = 3 \]

3. **Length along the z-axis (PE)**:
\[ PE = |z_1 - z_2| = |5 - (-1)| = |5 + 1| = |6| = 6 \]

### Step 3: Use the formula for the diagonal of the parallelepiped
The length of the diagonal \( D \) of a rectangular parallelepiped can be calculated using the formula:
\[ D = \sqrt{l^2 + b^2 + h^2} \]
where \( l \), \( b \), and \( h \) are the lengths of the edges along the x, y, and z axes respectively.

Substituting the values we calculated:
- \( l = 2 \)
- \( b = 3 \)
- \( h = 6 \)

We get:
\[ D = \sqrt{2^2 + 3^2 + 6^2} \]
Calculating this step-by-step:
\[ D = \sqrt{4 + 9 + 36} \]
\[ D = \sqrt{49} \]
\[ D = 7 \]

### Conclusion
The length of the diagonal of the parallelepiped is \( 7 \) units.

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