To determine the numerical length of \( NO \), we first need to express the relationship among the segments \( MN \), \( NO \), and \( MO \). Given:
- \( MO = 3x + 6 \)
- \( NO = 5x \)
- \( MN = x \)
Since point \( N \) divides line segment \( MO \) into two segments, we have:
\[ MO = MN + NO \]
Substituting the expressions we have into that equation:
\[ 3x + 6 = x + 5x \]
Simplifying the right side, we get:
\[ 3x + 6 = 6x \]
Now, to isolate \( x \), we can rearrange the equation:
\[ 3x + 6 - 6x = 0 \] \[ -3x + 6 = 0 \] \[ 6 = 3x \] \[ x = 2 \]
Next, we can substitute \( x = 2 \) into the expression for \( NO \):
\[ NO = 5x = 5(2) = 10 \]
Therefore, the numerical length of \( NO \) is:
\[ \boxed{10} \]
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