document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192419>Question 1192419</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Prove that if two intersecting line segments meet at their midpoints, the line<BR>\n" );
document.write( "segments joining their endpoints are congruent.<BR>\n" );
document.write( "Given: O is the midpoint of XY and of AB<BR>\n" );
document.write( "Prove: XA ≅ BY<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848993 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848993\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Definition of Midpoint:**\r<BR>\n" );
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document.write( "* Since O is the midpoint of XY, we know that OX ≅ OY.<BR>\n" );
document.write( "* Since O is the midpoint of AB, we know that OA ≅ OB.\r<BR>\n" );
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document.write( "**2. Vector Addition:**\r<BR>\n" );
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document.write( "* Express XA as the sum of vectors: XA = XO + OA <BR>\n" );
document.write( "* Express BY as the sum of vectors: BY = BO + OY\r<BR>\n" );
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document.write( "**3. Substitute with Equal Vectors**\r<BR>\n" );
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document.write( "* Since OX = -OY and OA = -OB, we can substitute:<BR>\n" );
document.write( "    * XA = -OY + OA<BR>\n" );
document.write( "    * BY = -OA + OY\r<BR>\n" );
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document.write( "**4. Rearrange**\r<BR>\n" );
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document.write( "* XA = -OY + OA <BR>\n" );
document.write( "* BY = OY - OA\r<BR>\n" );
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document.write( "* Observe that XA and BY are the same vector but with opposite signs.\r<BR>\n" );
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document.write( "**5. Conclusion**\r<BR>\n" );
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document.write( "* Since XA and BY have the same magnitude but opposite directions, they are congruent.\r<BR>\n" );
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document.write( "**Therefore, if two intersecting line segments meet at their midpoints, the line segments joining their endpoints are congruent.**\r<BR>\n" );
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document.write( "This proof utilizes vector concepts to demonstrate the congruence of the line segments.<BR>\n" );
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