document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1193334>Question 1193334</A>: <I><SPAN STYLE=\"word-break: break-all;\"> EF is the median of trapezoid ABCD in the figure below. Use the following theorems to answer the questions.<BR>\n" );
document.write( "If three (or more) parallel lines intercept congruent line segments on one transversal, then they intercept congruent line segments on any transversal.<BR>\n" );
document.write( "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.<BR>\n" );
document.write( "Suppose that AB = 11.4 and DC = 17.2.\r<BR>\n" );
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document.write( "Find MF.<BR>\n" );
document.write( "Find EM.<BR>\n" );
document.write( "Find EF.<BR>\n" );
document.write( "Find<BR>\n" );
document.write( "1/2(AB + DC).<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848617 by </B><A HREF=/tutors/aboutme.mpl?userid=proyaop><B>proyaop(69)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=proyaop><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848617\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> We are working with the median \( EF \) of trapezoid \( ABCD \), where:\r<BR>\n" );
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document.write( "- \( AB \) and \( DC \) are the parallel sides of the trapezoid,<BR>\n" );
document.write( "- The median \( EF \) is the line segment joining the midpoints of the non-parallel sides \( AD \) and \( BC \),<BR>\n" );
document.write( "- \( AB = 11.4 \) and \( DC = 17.2 \).\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### Theorems Used:\r<BR>\n" );
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document.write( "1. **The Length of the Median**:  <BR>\n" );
document.write( "   The median of a trapezoid is parallel to the bases and its length is the average of the lengths of the bases:\r<BR>\n" );
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document.write( "   \[<BR>\n" );
document.write( "   EF = \frac{AB + DC}{2}<BR>\n" );
document.write( "   \]\r<BR>\n" );
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document.write( "2. **Properties of Parallel Lines and Midpoints**:  <BR>\n" );
document.write( "   The points \( M \) and \( F \), the midpoints of \( AD \) and \( BC \), divide the sides of the trapezoid proportionally.\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### Step 1: Find \( EF \)\r<BR>\n" );
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document.write( "Using the formula for the median's length:\r<BR>\n" );
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document.write( "\[<BR>\n" );
document.write( "EF = \frac{AB + DC}{2}<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "Substitute the given values:\r<BR>\n" );
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document.write( "\[<BR>\n" );
document.write( "EF = \frac{11.4 + 17.2}{2} = \frac{28.6}{2} = 14.3<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "Thus, \( EF = 14.3 \).\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### Step 2: Find \( \frac{1}{2}(AB + DC) \)\r<BR>\n" );
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document.write( "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:\r<BR>\n" );
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document.write( "\[<BR>\n" );
document.write( "\frac{1}{2}(AB + DC) = 14.3<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### Step 3: Relationship Between \( MF \), \( EM \), and \( EF \)\r<BR>\n" );
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document.write( "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.\r<BR>\n" );
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document.write( "### Final Answers:<BR>\n" );
document.write( "1. \( EF = 14.3 \)<BR>\n" );
document.write( "2. \( \frac{1}{2}(AB + DC) = 14.3 \)<BR>\n" );
document.write( "3. \( MF \) and \( EM \) cannot be determined without more information about the geometry of \( AD \) and \( BC \). </SPAN>\n" );
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