document.write( "Question 748157: In a triangle, a segment is drawn joining the midpoint of two sides. What is true about this segment? \n" ); document.write( "
Algebra.Com's Answer #455347 by MathLover1(20850)![]() ![]() You can put this solution on YOUR website! The straight line connecting midpoints of two sides of a triangle is \n" ); document.write( "\n" ); document.write( "The straight line connecting midpoints of two sides of a triangle is parallel to the third side of the triangle. \r \n" ); document.write( "\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "This means that the angles ECD and DBF are congruent as the corresponding angles of these triangles. \n" ); document.write( "Hence, the straight lines AC and BF are parallel, because these angles are the alternate interior angles formed by the transverse line BC \n" ); document.write( "(see the lesson Parallel lines under the topic Angles, complementary, supplementary angles in the section Geometry in this site). \n" ); document.write( "This means also that the segments CE and BF are of equal length as the corresponding sides of triangles EDC and FDB. \n" ); document.write( "Since the point E is the midpoint of the side AB and the segments AE and CE are of equal length, this implies that the segments BF and AE are of equal length.\r \n" ); document.write( "\n" ); document.write( "Thus, we have proved that in the quadrilateral ADFE the two opposite sides BF and AE are parallel and have equal length.\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "At this point we can refer to the geometry fact proven in the lesson Properties of the sides of parallelograms (see the Theorem 1 of that lesson): \n" ); document.write( "if a quadrilateral has two opposite sides parallel and of equal length, then two other opposite sides of the quadrilateral are parallel and of equal length too.\r \n" ); document.write( "\n" ); document.write( "It implies that the straight lines AB and EF are parallel. \n" ); document.write( "This is exactly what we were going to prove.\r \n" ); document.write( " \n" ); document.write( " \n" ); document.write( "\n" ); document.write( " \n" ); document.write( " |