document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=823783>Question 823783</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The coordinates for a rhombus are given as(2a,0),(0,2b),(-2a,0),and (0,2b).Write a plan to prove that the midpoints of the sides of a rhombus determine a rectangle using coordinate geometry. Be sure to include the formulas.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #495914 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=495914\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-0.9%2C0.9%2C-0.9%2C0.9%2C%0D%0Agrid%280%29%2C%0D%0Aline%280.6%2C0%2C0%2C0.8%29%2Cline%280.6%2C0%2C0%2C-0.8%29%2C%0D%0Aline%28-0.6%2C0%2C0%2C0.8%29%2Cline%28-0.6%2C0%2C0%2C-0.8%29%2C%0D%0Alocate%280.58%2C0.11%2CA%282a%2C0%29%29%2Clocate%280.03%2C0.85%2CB%280%2C2b%29%29%2C%0D%0Alocate%28-0.54%2C0%2CC%28-2a%2C0%29%29%2Clocate%280.03%2C-0.77%2CD%280%2C-2b%29%29%2C%0D%0Ared%28circle%280.3%2C0.4%2C0.02%29%29%2Cred%28circle%280.3%2C-0.4%2C0.02%29%29%2C%0D%0Ared%28circle%28-0.3%2C0.4%2C0.02%29%29%2Cred%28circle%28-0.3%2C-0.4%2C0.02%29%29%2C%0D%0Alocate%280.33%2C0.45%2CE%29%2Clocate%280.33%2C-0.35%2CH%29%2C%0D%0Alocate%28-0.38%2C0.45%2CF%29%2Clocate%28-0.38%2C-0.35%2CG%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > E, F, G, and H are the midpoints of sides AB, BC, CD, and DA respectively.<BR>\n" );
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document.write( "Step 1: Determine the coordinates of the midpoints<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5BE%5D=%28x%5BA%5D%2Bx%5BB%5D%29%2F2=%282a%2B0%29%2F2=2a%2F2=a ALIGN=MIDDLE  ALT=\"x%5BE%5D=%28x%5BA%5D%2Bx%5BB%5D%29%2F2=%282a%2B0%29%2F2=2a%2F2=a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5BE%5D=%28x%5BA%5D%2Bx%5BB%5D%29%2F2=%280%2B2b%29%2F2=2b%2F2=b ALIGN=MIDDLE  ALT=\"y%5BE%5D=%28x%5BA%5D%2Bx%5BB%5D%29%2F2=%280%2B2b%29%2F2=2b%2F2=b\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and so on.<BR>\n" );
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document.write( "Step 2: Show that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5BE%5D=b=y%5BF%5D ALIGN=MIDDLE  ALT=\"y%5BE%5D=b=y%5BF%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > , which proves that <BR>\n" );
document.write( "EF is part of the horizontal line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=b ALIGN=MIDDLE  ALT=\"y=b\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Similarly prove that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5BF%5D=-a=x%5BG%5D ALIGN=MIDDLE  ALT=\"x%5BF%5D=-a=x%5BG%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > , which proves that<BR>\n" );
document.write( "FG is part of the vertical line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-a ALIGN=MIDDLE  ALT=\"x=-a\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "You can similarly prove that<BR>\n" );
document.write( "GH is part of the horizontal line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-b ALIGN=MIDDLE  ALT=\"y=-b\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and<BR>\n" );
document.write( "HE is part of the vertical line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=a ALIGN=MIDDLE  ALT=\"x=a\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
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document.write( "Step 3:<BR>\n" );
document.write( "State that<BR>\n" );
document.write( "two horizontal lines are parallel to each other, so EF is parallel to GH,<BR>\n" );
document.write( "and that<BR>\n" );
document.write( "two vertical lines are parallel to each other, so FG is parallel to HE.<BR>\n" );
document.write( "Then state that since opposite sides are parallel, quadrilateral EFGH is a parallelogram.<BR>\n" );
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document.write( "Step 4:<BR>\n" );
document.write( "State that horizontal lines are perpendicular to vertical lines, so<BR>\n" );
document.write( "EF is perpendicular to FG (meaning that angle EFG is a right angle),<BR>\n" );
document.write( "FG is perpendicular to GH (meaning that angle FGH is a right angle),<BR>\n" );
document.write( "GH is perpendicular to HE (meaning that angle GHE is a right angle), and<BR>\n" );
document.write( "HE is perpendicular to EF (meaning that angle HEF is a right angle).<BR>\n" );
document.write( "That means that in parallelogram EFGH,<BR>\n" );
document.write( "the angles at F, G, H, and E are right angles.<BR>\n" );
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document.write( "Step 5:<BR>\n" );
document.write( "State that since EFGH is a parallelogram with 4 right angles, it is a rectangle. </SPAN>\n" );
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