To find the length of the median of the trapezoid defined by the vertices \( F, U, G, E, V, D \) with the provided top and bottom bases, we will use the formula for the length of the median (or midsegment) of a trapezoid:
\[ m = \frac{b_1 + b_2}{2} \]
where \( b_1 \) is the length of the top base and \( b_2 \) is the length of the bottom base.
From the question, we have:
- The top base (length \( b_1 \)) is given as \( 2x + 4 \).
- The bottom base (length \( b_2 \)) is given as \( 2x + 18 \).
Now let’s calculate the length of the median \( m \):
\[ m = \frac{(2x + 4) + (2x + 18)}{2} \]
Now simplify the expression:
\[ m = \frac{2x + 4 + 2x + 18}{2} \] \[ m = \frac{4x + 22}{2} \] \[ m = 2x + 11 \]
Next, we also need to determine the value of \( x \) using the center length of the trapezoid which is given as \( 5x - 16 \). In a trapezoid, the median should also equal the average of the two bases. Thus, we set:
\[ 2x + 11 = 5x - 16 \]
Now we can solve for \( x \):
\[ 11 + 16 = 5x - 2x \] \[ 27 = 3x \] \[ x = 9 \]
Now substituting \( x = 9 \) back into the equation for median \( m \):
\[ m = 2(9) + 11 = 18 + 11 = 29 \]
Thus, the length of the median of the trapezoid is:
\[ \boxed{29} \]
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