document.write( "Question 161720: show that the quadrilateral formed by joining the mid-points of the sides of a square is also a square. \n" ); document.write( "

Algebra.Com's Answer #119194 by MathLover1(20850) $\"\"$ $\"About$

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Given:
\n" ); document.write( " $\"P\"$ , $\"Q\"$ , $\"R\"$ and $\"S+\"$ are the $\"mid-points+\"$ of $\"AB\"$ , $\"BC\"$ , $\"CD\"$ and $\"DA\"$ respectively.
\n" ); document.write( "To prove:
\n" ); document.write( " $\"PQRS\"$ is $\"a\"$ $\"rhombus\"$ . \r
\n" ); document.write( "\n" ); document.write( "Proof:\r
\n" ); document.write( "\n" ); document.write( "Consider triangle $\"BAC+\"$
\n" ); document.write( " $\"PQ\"$ || $\"AC\"$ and $\"PQ+=+%281%2F2%29AC\"$ Е.(1)\r
\n" ); document.write( "\n" ); document.write( "(In a triangle the segment joining the mid-points of
\n" ); document.write( "two sides are parallel and equal to third side)
\n" ); document.write( "Consider triangle $\"ADC+\"$ ,\r
\n" ); document.write( "\n" ); document.write( " $\"SR\"$ || $\"AC\"$ and $\"SR+=+%281%2F2%29AC\"$ Е.(2)\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "From (1) and (2),
\n" ); document.write( " $\"PQ\"$ || $\"SR\"$ and $\"PQ=SR\"$ \r
\n" ); document.write( "\n" ); document.write( " $\"PQRS\"$ is a parallelogram ЕЕЕЕЕЕ.(3)\r
\n" ); document.write( "\n" ); document.write( " $\"AD+=+BC\"$ Е.. (opposite sides of a rectange)\r
\n" ); document.write( "\n" ); document.write( "So
\n" ); document.write( " $\"%281%2F2%29AD+=%281%2F2%29BC\"$ Е..\r
\n" ); document.write( "\n" ); document.write( "i.e. $\"AS+=+BQ+\"$ \r
\n" ); document.write( "\n" ); document.write( "Consider triangle $\"+APS+\"$ , and triangle $\"+BPQ+\"$ ,\r
\n" ); document.write( "\n" ); document.write( " $\"AP+=+BP\"$ Е. ( $\"P\"$ is the mid-point of $\"AB\"$ )
\n" ); document.write( " $\"AS+=+BQ\"$ \r
\n" ); document.write( "\n" ); document.write( "angle $\"SAP\"$ = $\"PBQ\"$ = $\"90degrees\"$ \r
\n" ); document.write( "\n" ); document.write( "So, $\"triangle\"$ $\"+APS+\"$ is $\"congruent\"$ to triangle $\"+BPQ+\"$ .... ( $\"SAS\"$ congruency condition)
\n" ); document.write( "
\n" ); document.write( " $\"+PS+=+PQ\"$ ЕЕЕЕЕЕЕЕЕ..(4)
\n" ); document.write( "From (3) and (4),
\n" ); document.write( " $\"PQRS\"$ is a parallelogram in which $\"PS+=+PQ+\"$ .
\n" ); document.write( " $\"PQRS\"$ is a $\"rhombus\"$ .
\n" ); document.write( " \n" ); document.write( "

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