Let the lengths of the four sides of the quadrilateral be represented as four consecutive even integers. We can denote them as:
- \( x \): the first even integer
- \( x + 2 \): the second even integer
- \( x + 4 \): the third even integer
- \( x + 6 \): the fourth even integer
The formula for the perimeter \( P \) of the quadrilateral is the sum of its side lengths:
\[ P = x + (x + 2) + (x + 4) + (x + 6) \]
Combining the terms gives:
\[ P = 4x + 12 \]
We are told that the perimeter is 5252 meters, so we can set up the equation:
\[ 4x + 12 = 5252 \]
To find \( x \), we first subtract 12 from both sides:
\[ 4x = 5252 - 12 \] \[ 4x = 5240 \]
Next, divide both sides by 4:
\[ x = \frac{5240}{4} = 1310 \]
Now we can determine the lengths of all four sides:
- First side: \( x = 1310 \)
- Second side: \( x + 2 = 1312 \)
- Third side: \( x + 4 = 1314 \)
- Fourth side: \( x + 6 = 1316 \)
The longest of these four side lengths is:
\[ \text{Longest side} = x + 6 = 1316 \text{ meters} \]
Therefore, the value of the longest of the four side lengths is \( \boxed{1316} \).