document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1038606>Question 1038606</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In a Reflection,the image of the line y-2x=3 is the line 2y-x=9.Then What Is The Line Of Reflection? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #653501 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=653501\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ONE WAY TO THE ANSWER:<BR>\n" );
document.write( "The slope of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28y-2x=3%29 ALIGN=MIDDLE  ALT=\"red%28y-2x=3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28y=2x%2B3%29 ALIGN=MIDDLE  ALT=\"red%28y=2x%2B3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The slope of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%282y-x=9%29 ALIGN=MIDDLE  ALT=\"green%282y-x=9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%282y=x%2B9%29 ALIGN=MIDDLE  ALT=\"green%282y=x%2B9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%28y=%281%2F2%29x%2B9%2F2%29 ALIGN=MIDDLE  ALT=\"green%28y=%281%2F2%29x%2B9%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2 ALIGN=MIDDLE  ALT=\"1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That means the two lines intersect at some point <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P ALIGN=MIDDLE  ALT=\"P\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-2%2C8%2C0%2C10%2C%0D%0Ared%28line%28-2%2C-1%2C4%2C11%29%29%2C%0D%0Agreen%28line%28-3%2C3%2C9%2C9%29%29%2C%0D%0Alocate%281%2C5%2CP%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That intersection point is part of the Line of Reflection, and<BR>\n" );
document.write( "the Line of Reflection is the bisector of the angle formed by the two lines:<BR>\n" );
document.write( " <BR>\n" );
document.write( "Since the slopes of the lines are reciprocals of each other,<BR>\n" );
document.write( "the slope of the bisector of that angle is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "We can find the coordinates of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P ALIGN=MIDDLE  ALT=\"P\"   DISABLED_style= \"max-width:90%; height:auto;\" > by solving the system formed by the two equations:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28y-2x=3%2C2y-x=9%29 ALIGN=MIDDLE  ALT=\"system%28y-2x=3%2C2y-x=9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=1%2Cy=5%29 ALIGN=MIDDLE  ALT=\"system%28x=1%2Cy=5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So the lines intersect at point <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%281%2C5%29 ALIGN=MIDDLE  ALT=\"P%281%2C5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "Based on the coordinates of that point,<BR>\n" );
document.write( "the equation of the Line of Reflection in point-slope form is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-5=1%28x-1%29 ALIGN=MIDDLE  ALT=\"y-5=1%28x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28y-5=x-1%29 ALIGN=MIDDLE  ALT=\"highlight%28y-5=x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "In slope-intercept form, the equation of the Line of Reflection is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x-1%2B5 ALIGN=MIDDLE  ALT=\"y=x-1%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" > <---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28y=x%2B4%29 ALIGN=MIDDLE  ALT=\"highlight%28y=x%2B4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-2%2C8%2C-1%2C9%2Cgrid%281%29%2C%0D%0Ared%28line%28-2%2C-1%2C4%2C11%29%29%2C%0D%0Agreen%28line%28-3%2C3%2C9%2C9%29%29%2C%0D%0Alocate%281.1%2C5%2CP%29%2C%0D%0Aline%28-2%2C2%2C6%2C10%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "NOTES:<BR>\n" );
document.write( "Is that the way, you were expected to get to the answer?<BR>\n" );
document.write( "Are you expected to name other concepts like \"translation\", \"reflection\", \"vectors\", \"matrices\" to \"show your work\"?<BR>\n" );
document.write( "Was a different way to the solution expected?<BR>\n" );
document.write( " <BR>\n" );
document.write( "Not convinced that the slope of the Line of Reflection is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ?<BR>\n" );
document.write( "Here is a concrete example, showing why the slope of the Line of Reflection is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "If the two lines formed the legs of an isosceles triangle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ABP ALIGN=MIDDLE  ALT=\"ABP\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "altitude <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=MP ALIGN=MIDDLE  ALT=\"MP\"   DISABLED_style= \"max-width:90%; height:auto;\" > of that triangle would be part of the bisector of angle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ABP ALIGN=MIDDLE  ALT=\"ABP\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "which is the Line of Reflection we are looking for.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-2%2C8%2C0%2C10%2C%0D%0Atriangle%281%2C5%2C3%2C9%2C5%2C7%29%2C%0D%0Aline%281%2C5%2C4%2C8%29%2C%0D%0Ared%28line%28-2%2C-1%2C4%2C11%29%29%2C%0D%0Agreen%28line%28-3%2C3%2C9%2C9%29%29%2C%0D%0Alocate%281%2C5%2CP%29%2C%0D%0Alocate%282.6%2C9.5%2CA%29%2C%0D%0Alocate%285%2C7%2CB%29%2C%0D%0Alocate%284%2C8.57%2CM%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M ALIGN=MIDDLE  ALT=\"M\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the midpoint of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AB ALIGN=MIDDLE  ALT=\"AB\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "How can we find a pair of points <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B ALIGN=MIDDLE  ALT=\"B\"   DISABLED_style= \"max-width:90%; height:auto;\" > to make such an isosceles triangle?<BR>\n" );
document.write( "Since the slope of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-2x=3 ALIGN=MIDDLE  ALT=\"y-2x=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "we get the coordinates of another point on line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-2x=3 ALIGN=MIDDLE  ALT=\"y-2x=3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "by adding to the coordinates of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%5BP%5D%2Cy%5BP%5D%29 ALIGN=MIDDLE  ALT=\"P%28x%5BP%5D%2Cy%5BP%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (1,2), or (2,4),  or (3,6), etc.<BR>\n" );
document.write( "Let us say that adding (2,4) to the coordinates of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P ALIGN=MIDDLE  ALT=\"P\"   DISABLED_style= \"max-width:90%; height:auto;\" > , we get point <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%28x%5Bp%5D%2B2%2Cy%5BP%5D%2B4%29 ALIGN=MIDDLE  ALT=\"A%28x%5Bp%5D%2B2%2Cy%5BP%5D%2B4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "at a distance <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282%5E2%2B4%5E2%29=sqrt%284%2B16%29=sqrt%2820%29 ALIGN=MIDDLE  ALT=\"sqrt%282%5E2%2B4%5E2%29=sqrt%284%2B16%29=sqrt%2820%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > from the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P ALIGN=MIDDLE  ALT=\"P\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Likewise, since the slope of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2y-x=9 ALIGN=MIDDLE  ALT=\"2y-x=9\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2 ALIGN=MIDDLE  ALT=\"1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > adding (4,2) to the coordinates of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P ALIGN=MIDDLE  ALT=\"P\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "we get point <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B ALIGN=MIDDLE  ALT=\"B\"   DISABLED_style= \"max-width:90%; height:auto;\" > on line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2y-x=9 ALIGN=MIDDLE  ALT=\"2y-x=9\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "at the same <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2820%29 ALIGN=MIDDLE  ALT=\"sqrt%2820%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > distance from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P ALIGN=MIDDLE  ALT=\"P\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "To get the coordinates of midpoint <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M ALIGN=MIDDLE  ALT=\"M\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "we average the coordinates of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B ALIGN=MIDDLE  ALT=\"B\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "So to get to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M ALIGN=MIDDLE  ALT=\"M\"   DISABLED_style= \"max-width:90%; height:auto;\" > from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P ALIGN=MIDDLE  ALT=\"P\"   DISABLED_style= \"max-width:90%; height:auto;\" > , we add<BR>\n" );
document.write( "\"rise\" <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284%2B2%29%2F2=6%2F2=3 ALIGN=MIDDLE  ALT=\"%284%2B2%29%2F2=6%2F2=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5BP%5D ALIGN=MIDDLE  ALT=\"y%5BP%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and<BR>\n" );
document.write( "\"run\" <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2B4%29%2F2=6%2F2=3 ALIGN=MIDDLE  ALT=\"%282%2B4%29%2F2=6%2F2=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5BP%5D ALIGN=MIDDLE  ALT=\"x%5BP%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >  .<BR>\n" );
document.write( "Since the\"rise\" and \"run\" changes in x- and y-coordinates are the same,<BR>\n" );
document.write( "the slope of the Line of Reflection is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );