document.write( "Question 1182583: The point P has coordinates (a,b). Another point Q is formed by reversing the coordinates of P, i.e. Q has coordinates (b,a).
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\n" ); document.write( "(ii) Show that the midpoint, M, of PQ lies on y=x. \n" ); document.write( "

Algebra.Com's Answer #812657 by ikleyn(52800) $\"\"$ $\"About$

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\n" ); document.write( "The point P has coordinates (a,b). Another point Q is formed by reversing the coordinates of P,
\n" ); document.write( "i.e. Q has coordinates (b,a).
\n" ); document.write( "(i) Show that PQ is perpendicular to the line y=x.
\n" ); document.write( "(ii) Show that the midpoint, M, of PQ lies on y=x.
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\n" ); document.write( "\n" ); document.write( "            In this problem, it is assumed that a =/= b --- otherwise, the statement loses sense.\r
\n" ); document.write( "\n" ); document.write( "            So, I will assume that a =/= b, although the problem does not says it explicitly.\r
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\n" ); document.write( "\n" ); document.write( "                        Part ( i )\r
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document.write( "Let's compare slopes of the line y = x  and the line (segment) PQ.\r\n" );
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document.write( "The line y = x has the slope equal to 1, OBVIOUSLY.\r\n" );
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document.write( "The segment PQ hs the slope  m =  =  = -1.\r\n" );
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document.write( "The slopes 1 and -1 are negatively reciprocal: 1*(-1) = -1; THEREFORE, the line y = x and the segment (the line) PQ are perpendicular.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Thus the first statement is proved.\r
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\n" ); document.write( "\n" ); document.write( " Part ( ii )\r
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document.write( "The coordinates of the midpoint are  (,).\r\n" );
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document.write( "Both x- and y- coordinates are equal.\r\n" );
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document.write( "It proves, that the midpoint lies on the line y = x.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Thus the second statement is proved, too.\r
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\n" ); document.write( "\n" ); document.write( "The points P and Q are mirror reflections each other about the line y = x.\r
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