document.write( "Question 1025719: A and B are on the same side of a line L. AD and BE are perpendiculars drawn to the line L. If C is the midpoint of AB then prove that CD = CE. \n" ); document.write( "

Algebra.Com's Answer #641103 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "A and B are on the same side of a line L. AD and BE are perpendiculars drawn to the line L.
\n" ); document.write( "If C is the midpoint of AB then prove that CD = CE.
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document.write( "The plot for this problem is shown in the Figure on the right.\r\n" );
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document.write( "We need to prove that two red colored segments are congruent.\r\n" );
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document.write( "Let us draw the perpendicular CF (green line) from the point C to the line DE.\r\n" );
document.write( "The perpendicular CF is parallel to both lines AD and BE.\r\n" );
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document.write( "Thus we have three parallel lines AD, CF and BE, and two transverses DE and AB.\r\n" );
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document.write( "Now apply this property:\r\n" );
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document.write( "    If three parallel lines cut off two congruent segments in one transverse line, \r\n" );
document.write( "    then they cut off two congruent segments in any other transverse line.\r\n" );
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document.write( "(See the lesson \r\n" );
document.write( "Three parallel lines cutting off congruent segments in a transverse line \r\n" );
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document.write( "                     Figure\r\n" );
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document.write( "Since the segments AC and CB are congruent, the segments DF and FE are congruent too, according to this property.\r\n" );
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document.write( "Now we have two right-angled triangles DCF and ECF.\r\n" );
document.write( "They have congruent legs DF and EF and the common leg CF.\r\n" );
document.write( "Hence, these triangles are congruent according to SAS-test for triangles congruency.\r\n" );
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document.write( "It implies that CD = CE as the corresponding sides of congruent triangles.\r\n" );
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document.write( "The proof is completed.\r\n" );
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