document.write( "Question 1156832: The midline of a trapezoid is not concurrent with the diagonals. Explain why. \n" ); document.write( "

Algebra.Com's Answer #779655 by ikleyn(52800) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( "            Tutor @greenestamps incorrectly interprets the notion of concurrency for straight lines.\r
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\n" ); document.write( "\n" ); document.write( "            Therefore, his solution is incorrect.\r
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document.write( "Let me start reminding you two facts from Geometry.\r\n" );
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document.write( "   1)  A trapezoid is a quadrilateral, which has two opposite sides PARALLEL and IS NOT a parallelogram, at the same time.\r\n" );
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document.write( "   2)  Three lines in a plane are called concurrent if they intersect in one common point.\r\n" );
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document.write( "So, the question is to explain why two diagonals of the trapezoid and its midline DO NOT intersect in one common point.\r\n" );
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document.write( "I will try to make my explanation as simple as possible.\r\n" );
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document.write( "Let the trapezoid be ABCD, with the bases a = AB and b = CD.\r\n" );
document.write( "Let O be the intersection point of diagonals.\r\n" );
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document.write( "Then the triangles AOB and DOC are similar. It is an elementary statement, so let take it with no proof.\r\n" );
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document.write( "The bases of these triangles, AB and DC, are in proportion   = .\r\n" );
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document.write( "Next, since a trapezoid is not a parallelogram, a =/= b; therefore  is not equal to 1.\r\n" );
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document.write( "It means that in triangles AOB and DOC the altitudes OE and OF, drawn from the point O to their bases AB and CD, \r\n" );
document.write( "are of different lengths.\r\n" );
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document.write( "From the other side, the midline of the trapezoid ABSD, cuts the altitude EF of the trapezoid in two congruent pieces.\r\n" );
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document.write( "    ( It is true for any trapezoid (!) )\r\n" );
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document.write( "This contradiction explains, why the bases of a trapezoid and its midline are not concurrent.\r\n" );
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