document.write( "Question 1067733: ABC is an isosceles triangle in which mAB =mAC.CD is the perpendicular drawn from C to the opposite side .Prove that (mBC)^2=2 (mAB).(mBD). \n" ); document.write( "

Algebra.Com's Answer #682919 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "ABC is an isosceles triangle in which mAB =mAC. CD is the perpendicular drawn from C to the opposite side.
\n" ); document.write( "Prove that (mBC)^2=2 (mAB).(mBD).
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document.write( "1.  Make a sketch.\r\n" );
document.write( "    Draw the triangle ABC and the perpendicular CD.\r\n" );
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document.write( "    Draw the median AE from the vertex A to the base BC.\r\n" );
document.write( "    Notice that the median AE is the altitude of the triangle ABC at the same time.\r\n" );
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document.write( "2.  The right-angle triangles BCD and ABE are SIMILAR.\r\n" );
document.write( "        ( Because they have the common acute angle B. \r\n" );
document.write( "          For right-angled triangles having a common acute angle, it is enough to be similar ! )\r\n" );
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document.write( "3.  From the similarity, you have this proportion for the measures of corresponding sides:\r\n" );
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document.write( "     = .      (1)\r\n" );
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document.write( "4.  Now notice that BE = .\r\n" );
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document.write( "    Together with (1), it implies that  = 2*|BD|*|AB|. \r\n" );
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document.write( "    QED.\r\n" );
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