Question 1193334: EF is the median of trapezoid ABCD in the figure below. Use the following theorems to answer the questions.
If three (or more) parallel lines intercept congruent line segments on one transversal, then they intercept congruent line segments on any transversal.
The line segment that joins the midpoints of two sides of a triangle is parallel to the third side and has a length equal to one-half the length of the third side.
Suppose that AB = 11.4 and DC = 17.2.
Find MF.
Find EM.
Find EF.
Find
1/2(AB + DC).
Answer by proyaop(69)   (Show Source):
You can put this solution on YOUR website! 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 \( DC = 17.2 \).
---
### 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.
---
### 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, \( EF = 14.3 \).
---
### 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:
\[
\frac{1}{2}(AB + DC) = 14.3
\]
---
### 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 \( BC \).
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