Question 1193334
We are working with the median \( EF \) of trapezoid \( ABCD \), where:

- \( AB \) and \( DC \) are the parallel sides of the trapezoid,
- The median \( EF \) is the line segment joining the midpoints of the non-parallel sides \( AD \) and \( BC \),
- \( AB = 11.4 \) and \( DC = 17.2 \).

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### Theorems Used:

1. **The Length of the Median**:  
   The median of a trapezoid is parallel to the bases and its length is the average of the lengths of the bases:

   \[
   EF = \frac{AB + DC}{2}
   \]

2. **Properties of Parallel Lines and Midpoints**:  
   The points \( M \) and \( F \), the midpoints of \( AD \) and \( BC \), divide the sides of the trapezoid proportionally.

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### Step 1: Find \( EF \)

Using the formula for the median's length:

\[
EF = \frac{AB + DC}{2}
\]

Substitute the given values:

\[
EF = \frac{11.4 + 17.2}{2} = \frac{28.6}{2} = 14.3
\]

Thus, \( EF = 14.3 \).

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### Step 2: Find \( \frac{1}{2}(AB + DC) \)

The value of \( \frac{1}{2}(AB + DC) \) is identical to the value of \( EF \), since the median's length is calculated as the average of the bases:

\[
\frac{1}{2}(AB + DC) = 14.3
\]

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### Step 3: Relationship Between \( MF \), \( EM \), and \( EF \)

Because \( M \) and \( F \) are the midpoints of \( AD \) and \( BC \), the line segment \( EF \) is split proportionally. However, without additional information about the non-parallel sides (\( AD \) and \( BC \)) or specific distances, \( MF \) and \( EM \) cannot be determined numerically.

### Final Answers:
1. \( EF = 14.3 \)
2. \( \frac{1}{2}(AB + DC) = 14.3 \)
3. \( MF \) and \( EM \) cannot be determined without more information about the geometry of \( AD \) and \( BC \).