document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=938100>Question 938100</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Two vertices of an equilateral triangle are (3,4)and (-2,3) the third vertex can be </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #571583 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=571583\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> We know one side of the triangle, connecting vertices <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%283%2C4%29 ALIGN=MIDDLE  ALT=\"A%283%2C4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%28-2%2C3%29 ALIGN=MIDDLE  ALT=\"B%28-2%2C3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The length of  side AB is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%283-%28-2%29%29%5E2%2B%284-3%29%5E2%29=sqrt%28%285%29%5E2%2B%281%29%5E2%29=sqrt%2825%2B1%29=sqrt%2826%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Side AB is a segment of a line with slope<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=%284-3%29%2F%283-%28-2%29%29=1%2F5 ALIGN=MIDDLE  ALT=\"m=%284-3%29%2F%283-%28-2%29%29=1%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "The midpoint of AB, point P, has coordinates<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5BP%5D=%283%2B%28-2%29%29%2F2=1%2F2 ALIGN=MIDDLE  ALT=\"x%5BP%5D=%283%2B%28-2%29%29%2F2=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5BP%5D=%284%2B3%29%2F2=7%2F2 ALIGN=MIDDLE  ALT=\"y%5BP%5D=%284%2B3%29%2F2=7%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The equilateral triangles with side AB are ABD and ABC,<BR>\n" );
document.write( "with C and D on opposite sides of segment AB.<BR>\n" );
document.write( "Visualizing the location of vertices C and D is easy,<BR>\n" );
document.write( "but I could not think of a simple way to calculate their coordinates.<BR>\n" );
document.write( " <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28210%2C300%2C-3%2C4%2C-1%2C9%2C%0D%0Agrid%280%29%2Clocate%280.6%2C3.5%2CP%29%2C%0D%0Agreen%28triangle%283%2C4%2C1.366%2C-0.83%2C-0.366%2C7.83%29%29%2C%0D%0Agreen%28triangle%28-2%2C3%2C-0.366%2C7.83%2C1.366%2C-0.83%29%29%2C%0D%0Agreen%28line%280.4%2C4%2C0.9%2C4.1%29%29%2Cgreen%28line%281%2C3.6%2C0.9%2C4.1%29%29%2C%0D%0Aline%283%2C4%2C-2%2C3%29%2Clocate%283%2C4%2CA%29%2Clocate%28-2.3%2C3%2CB%29%2C%0D%0Alocate%281.5%2C-0.5%2CC%29%2Clocate%28-0.7%2C8%2CD%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The perpendicular bisector of AB contains the altitude of those triangles.<BR>\n" );
document.write( "That perpendicular bisector is a line perpendicular to AB,<BR>\n" );
document.write( "and passing through point P (the midpoint of AB).<BR>\n" );
document.write( "Being perpendicular to a line with slope <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=1%2F5 ALIGN=MIDDLE  ALT=\"m=1%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "that altitude is part of a line with slope <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2Fm=-5 ALIGN=MIDDLE  ALT=\"-1%2Fm=-5\"   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=PA=%281%2F2%29%2AAB=sqrt%2826%29%2F2 ALIGN=MIDDLE  ALT=\"PA=%281%2F2%29%2AAB=sqrt%2826%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Since angle PAC is a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=60%5Eo ALIGN=MIDDLE  ALT=\"60%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28PAC%29=PC%2FPA=sqrt%283%29 ALIGN=MIDDLE  ALT=\"tan%28PAC%29=PC%2FPA=sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PC=sqrt%283%29%2APA=sqrt%283%29%2Asqrt%2826%29%2F2=sqrt%2878%29%2F2 ALIGN=MIDDLE  ALT=\"PC=sqrt%283%29%2APA=sqrt%283%29%2Asqrt%2826%29%2F2=sqrt%2878%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So the length of PC and PD is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PC=PC=sqrt%2878%29%2F2 ALIGN=MIDDLE  ALT=\"PC=PC=sqrt%2878%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "but what are the coordinates of C and D?<BR>\n" );
document.write( " <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28210%2C390%2C-3%2C4%2C-2%2C11%2C%0D%0Alocate%280.6%2C3.5%2CP%29%2Cgreen%28arrow%280.5%2C-2%2C0.5%2C15%29%29%2C%0D%0Alocate%280.6%2C10.8%2Cgreen%28x=1%2F2%29%29%2C%0D%0Agreen%28triangle%280.5%2C3.5%2C1.366%2C-0.83%2C0.5%2C-0.83%29%29%2C%0D%0Agreen%28triangle%280.5%2C3.5%2C0.5%2C7.83%2C-0.366%2C7.83%29%29%2C%0D%0Agreen%28rectangle%280.5%2C-0.83%2C0.7%2C-0.63%29%29%2C%0D%0Agreen%28rectangle%280.5%2C7.83%2C0.3%2C7.63%29%29%2C%0D%0Aline%283%2C4%2C-2%2C3%29%2Clocate%283%2C4%2CA%29%2Clocate%28-2.3%2C3%2CB%29%2C%0D%0Alocate%281.5%2C-0.5%2CC%29%2Clocate%28-0.7%2C8%2CD%29%2C%0D%0Alocate%28-0.1%2C8.3%2Cgreen%28a%29%29%2Clocate%280.8%2C-0.7%2Cgreen%28a%29%29%2C%0D%0Alocate%280.6%2C5.9%2Cgreen%28b%29%29%2Clocate%280.6%2C1.4%2Cgreen%28b%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Since the slope of CD is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5 ALIGN=MIDDLE  ALT=\"-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=5a ALIGN=MIDDLE  ALT=\"b=5a\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and in those two green right triangles<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2%2Bb%5E2=a%5E2%2B%285a%29%5E2=a%5E2%2B25a%5E2=26a%5E2=%28sqrt%2878%29%2F2%29%5E2=78%2F4 ALIGN=MIDDLE  ALT=\"a%5E2%2Bb%5E2=a%5E2%2B%285a%29%5E2=a%5E2%2B25a%5E2=26a%5E2=%28sqrt%2878%29%2F2%29%5E2=78%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2=78%2F4%2A26=3%2F4 ALIGN=MIDDLE  ALT=\"a%5E2=78%2F4%2A26=3%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=sqrt%283%29%2F2 ALIGN=MIDDLE  ALT=\"a=sqrt%283%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=5sqrt%283%29%2F2 ALIGN=MIDDLE  ALT=\"b=5sqrt%283%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Then,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5BC%5D=1%2F2%2Bsqrt%283%29%2F2=%281%2Bsqrt%283%29%29%2F2=about1.37 ALIGN=MIDDLE  ALT=\"x%5BC%5D=1%2F2%2Bsqrt%283%29%2F2=%281%2Bsqrt%283%29%29%2F2=about1.37\"   DISABLED_style= \"max-width:90%; height:auto;\" > (rounded),<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5BC%5D=7%2F2-5sqrt%283%29%2F2=%287-5sqrt%283%29%29%2F2=about-0.83 ALIGN=MIDDLE  ALT=\"y%5BC%5D=7%2F2-5sqrt%283%29%2F2=%287-5sqrt%283%29%29%2F2=about-0.83\"   DISABLED_style= \"max-width:90%; height:auto;\" > (rounded),<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5BD%5D=1%2F2-sqrt%283%29%2F2=%281-sqrt%283%29%29%2F2=about-0.37 ALIGN=MIDDLE  ALT=\"x%5BD%5D=1%2F2-sqrt%283%29%2F2=%281-sqrt%283%29%29%2F2=about-0.37\"   DISABLED_style= \"max-width:90%; height:auto;\" > (rounded), and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5BD%5D=7%2F2%2B5sqrt%283%29%2F2=%287%2B5sqrt%283%29%29%2F2=about7.83 ALIGN=MIDDLE  ALT=\"y%5BD%5D=7%2F2%2B5sqrt%283%29%2F2=%287%2B5sqrt%283%29%29%2F2=about7.83\"   DISABLED_style= \"max-width:90%; height:auto;\" > (rounded). </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );