document.write( "Question 1091561: Prove that the midpoint of a kite form a rectangle
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Algebra.Com's Answer #705975 by ikleyn(52803) $\"\"$ $\"About$

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document.write( "1.  Use the fact that the mid-line of a triangle \r\n" );
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document.write( "        a) is parallel to the base of the triangle and \r\n" );
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document.write( "        b) has the length half of that of the base.\r\n" );
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document.write( "           See the lesson  The line segment joining the midpoints of two sides of a triangle  in this site.  \r\n" );
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document.write( "    In this way, you prove that the quadrilateral formed by the side midpoints of the kite has opposite sides parallel and congruent.\r\n" );
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document.write( "    It implies that this quadrilateral is a parallelogram.\r\n" );
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document.write( "2.  Now you need to prove that the diagonals of a kite are perpendicular.\r\n" );
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document.write( "    Draw these diagonals.\r\n" );
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document.write( "         a) notice that one diagonal divides the kite in two congruent triangles\r\n" );
document.write( "            (according to SSS-congruency test for triangles).\r\n" );
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document.write( "            Hence, the common side, which is the diagonal, is at the same time the bisector of the two kite's angles.\r\n" );
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document.write( "         b) From the other side, the other diagonal cuts the kite in two isosceles triangles.\r\n" );
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document.write( "            Since the first diagonal is the angle bisector of these triangles, it is the altitude at the same time.\r\n" );
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document.write( "            Thus we proved that the diagonals of a kite are perpendicular.\r\n" );
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document.write( "3.  It implies that the parallelogram under the question is a rectangle.\r\n" );
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