document.write( "Question 1022906: Prove that if the diagonals of a parallegram are equal it is a rectangle. \n" ); document.write( "

Algebra.Com's Answer #638465 by robertb(5830) $\"\"$ $\"About$

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Consider the parallelogram ABCD on the Cartesian plane, with coordinates A(0,0), B(a,0), C(c+a,d), and D(c,d). Without loss of generality, we can assume a>0 and $\"c%3E=0\"$ . (We wish to show that c = 0.)
\n" ); document.write( "The two diagonals are AC and BD.\r
\n" ); document.write( "\n" ); document.write( "Now apply the distance formula to the endpoints of AC and BD:\r
\n" ); document.write( "\n" ); document.write( " $\"abs%28AC%29%5E2+=+abs%28BD%29%5E2\"$
\n" ); document.write( "==> $\"%28c%2Ba%29%5E2+%2B+d%5E2+=+%28c-a%29%5E2%2Bd%5E2\"$ \r
\n" ); document.write( "\n" ); document.write( "==> $\"c%5E2+%2B2ac%2Ba%5E2+%2Bd%5E2+=+c%5E2+-2ac%2Ba%5E2+%2B+d%5E2\"$ , after expansion;\r
\n" ); document.write( "\n" ); document.write( "==> 4ac = 0, after simplification.\r
\n" ); document.write( "\n" ); document.write( "Since a>0, we then have c = 0. (This implies that angle DAB and angle ABC are right angles.)\r
\n" ); document.write( "\n" ); document.write( "Therefore parallelogram ABCD is a rectangle. \n" ); document.write( "

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