document.write( "Question 999912: Show that the medians of two similar triangles are in the ratio the same as the ratio of their corresponding sides. \n" ); document.write( "

Algebra.Com's Answer #617454 by ikleyn(52803) $\"\"$ $\"About$

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document.write( "Let two triangles    and    are similar,  so that the pairs of their corresponding sides    and  , \r\n" );
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document.write( "   and  ,  and    and    are proportional with the same  (common)  coefficient of proportionality.\r\n" );
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document.write( "Now consider two corresponding medians    and  .  Consider the triangles    and  . \r\n" );
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document.write( "They have two pairs of proportional sides    and  ,    and    with the same coefficient proportionality. \r\n" );
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document.write( "For the last pair,    and  ,  it is true because these segments are halves of the corresponding sides    and  .\r\n" );
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document.write( "The angles L and L  between these proportional sides are congruent,  as they are corresponding angles of the similar original triangles. \r\n" );
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document.write( "Thus the triangles    and    have two pairs of proportional sides and the congruent angles between them. \r\n" );
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document.write( "According to the  SAS-test of similarity for triangles,  these triangles are similar.\r\n" );
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document.write( "Therefore,  their sides    and    are proportional with the same coefficient of proportionality. \r\n" );
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document.write( "It is exactly what has to be proved,  since    and    are the corresponding medians of the original triangles. \r\n" );
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