In a parallelogram, the diagonals bisect each other. This means that the length of diagonal KM is divided into two equal segments, which are KN and NM.
Given:
- KM = 5x + 1
- NM = 2x + 9
Since N is the midpoint of KM, we have: \[ KN = NM \]
Thus, we can set up the equation: \[ \frac{KM}{2} = NM \]
This can be rewritten as: \[ \frac{5x + 1}{2} = 2x + 9 \]
To eliminate the fraction, multiply both sides by 2: \[ 5x + 1 = 4x + 18 \]
Now, isolate x by subtracting 4x from both sides: \[ 5x - 4x + 1 = 18 \]
This simplifies to: \[ x + 1 = 18 \]
Subtract 1 from both sides: \[ x = 17 \]
Therefore, the value of \( x \) is \( \boxed{17} \).