document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=959180>Question 959180</A>: <I><SPAN STYLE=\"word-break: break-all;\"> OPQR is a parallelogram where O is the origin (0,0).<BR>\n" );
document.write( "The co-ordinates of a point P are (k,0) and the co-ordinates of a point Q are (4,2).<BR>\n" );
document.write( "The point R lies on the line y = 2x.\r<BR>\n" );
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document.write( "How do I find the value of k?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #586286 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=586286\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> OPQR is a parallelogram.\r<BR>\n" );
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document.write( "point O is at (0,0).<BR>\n" );
document.write( "point P is at (k,0)<BR>\n" );
document.write( "point Q is at (4,2)<BR>\n" );
document.write( "point R lies on the line y = 2x.\r<BR>\n" );
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document.write( "if you draw this parallelogram on a graph:\r<BR>\n" );
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document.write( "point R would be top left.<BR>\n" );
document.write( "point Q would be top right.<BR>\n" );
document.write( "point P would be bottom right.<BR>\n" );
document.write( "point O would be bottom left.\r<BR>\n" );
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document.write( "reading clockwise from top left, the parallelogram would be RQPO.\r<BR>\n" );
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document.write( "since it is a parallelogram, opposite sides are parallel.\r<BR>\n" );
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document.write( "this means that OR is parallel to PQ and RQ is parallel to OP.\r<BR>\n" );
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document.write( "OP is on the x-axis since it goes from (0,0) to (k,0).\r<BR>\n" );
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document.write( "the fact that the y-coordinate of both points is at y = 0 tells you that.\r<BR>\n" );
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document.write( "the implicit assumption is that the line y = 2x forms one of the sides of the parallelogram.\r<BR>\n" );
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document.write( "if the line y = 2x connects through point R, this means that point O must be on that line as well.\r<BR>\n" );
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document.write( "this means the line segment OR is one of the sides of the parallelogram.\r<BR>\n" );
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document.write( "the opposite side to that is the line segment PQ.\r<BR>\n" );
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document.write( "since those two line segments need to be parallel, this means the slope of the line segment PQ must also be equal to 2.\r<BR>\n" );
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document.write( "since slope is equal to (y2-y1) / (x2-x1), then you can solve for k.\r<BR>\n" );
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document.write( "slope of line segment PQ goes from (k,0) to (4,2)\r<BR>\n" );
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document.write( "y2-y1 is equal to 2.\r<BR>\n" );
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document.write( "in order for the slope to be equal to 2, 4-k must be equal to 1.\r<BR>\n" );
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document.write( "this occurs when k = 3.\r<BR>\n" );
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document.write( "that's your solution.\r<BR>\n" );
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document.write( "you can also solve for point R, although you don't need to.\r<BR>\n" );
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document.write( "since point R is on the line segment RQ, and since the line segment RQ must be parallel to the x-axis because the line segment OP is on the x-axis, then point R must have a y value of 2.\r<BR>\n" );
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document.write( "slope of line PQ is therefore equal to (y2 - y1) / (x2 - x1) = (2-0) / (?-0).\r<BR>\n" );
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document.write( "since 2-0 is equal to 2, then ?-0 has to be equal to 1, so point R is at (1,0).\r<BR>\n" );
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document.write( "the opposite sides of a parallelogram are also equal in length.\r<BR>\n" );
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document.write( "the length of line segments RQ and OP is 3.\r<BR>\n" );
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document.write( "the length of line segments OR and PQ is sqrt(2^2 + 1^2) which is equal to sqrt(5).\r<BR>\n" );
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document.write( "here's a picture of your parallelogram.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2015/032802.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( " </SPAN>\n" );
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