document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=477701>Question 477701</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Two vertices of an equilateral triangle are (4,0) and (-6,0). find the coordinates of the point(s) that represent the 3rd vertex. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #327352 by </B><A HREF=/tutors/aboutme.mpl?userid=Mathmama><B>Mathmama(2)</B></A><img src=\"/images/stars/iconNewId_16x16.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Mathmama><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=327352\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> First in an equilateral triangle all the sides are the same length\r<BR>\n" );
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document.write( "Let's call the triangle ABC  where A(4,0)  B(-6,0)  and C (x,y)\r<BR>\n" );
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document.write( "the sides of the triangle are AB, BC and AC<BR>\n" );
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document.write( "To find the length of a side of the Triangle we need to use the formula to calculate the distance between two points (x1, y1) and (x2, y2)\r<BR>\n" );
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document.write( "    Distance = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%28x%5B1%5D+-x%5B2%5D%29%5E2++%2B+%28y%5B1%5D+-+y%5B2%5D%29%5E2%29 ALIGN=MIDDLE  ALT=\"sqrt%28%28x%5B1%5D+-x%5B2%5D%29%5E2++%2B+%28y%5B1%5D+-+y%5B2%5D%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Distance AB = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%284+-+%28-6%29%29%5E2+%2B+%280+-+0%29%5E2%29 ALIGN=MIDDLE  ALT=\"sqrt%28%284+-+%28-6%29%29%5E2+%2B+%280+-+0%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "            = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%284+%2B+6%29%5E2%29+%2B+0%29 ALIGN=MIDDLE  ALT=\"sqrt%28%284+%2B+6%29%5E2%29+%2B+0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "            =<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2810%5E2%29 ALIGN=MIDDLE  ALT=\"sqrt%2810%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 10\r<BR>\n" );
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document.write( "Since all sides are equal then length of BC = 10 and length of AC is also 10\r<BR>\n" );
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document.write( "Distance BC = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%28-6-x%29%5E2%2B%280+-+y%29%5E2%29 ALIGN=MIDDLE  ALT=\"sqrt%28%28-6-x%29%5E2%2B%280+-+y%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "             = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%28-6-x%29%5E2+%2B+y%5E2%29 ALIGN=MIDDLE  ALT=\"sqrt%28%28-6-x%29%5E2+%2B+y%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Distance AC = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%284-x%29%5E2%2B%280+-+y%29%5E2%29 ALIGN=MIDDLE  ALT=\"sqrt%28%284-x%29%5E2%2B%280+-+y%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "             = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%284-x%29%5E2+%2B+y%5E2%29 ALIGN=MIDDLE  ALT=\"sqrt%28%284-x%29%5E2+%2B+y%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "BC = AC<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%28-6-x%29%5E2+%2B+y%5E2%29 ALIGN=MIDDLE  ALT=\"sqrt%28%28-6-x%29%5E2+%2B+y%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  =   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%284-x%29%5E2+%2B+y%5E2%29 ALIGN=MIDDLE  ALT=\"sqrt%28%284-x%29%5E2+%2B+y%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "If we square both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-6-x%29%5E2+%2B+y%5E2 ALIGN=MIDDLE  ALT=\"%28-6-x%29%5E2+%2B+y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >  =   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284-x%29%5E2+%2B+y%5E2 ALIGN=MIDDLE  ALT=\"%284-x%29%5E2+%2B+y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2 ALIGN=MIDDLE  ALT=\"y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides \r<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-6-x%29%5E2+ ALIGN=MIDDLE  ALT=\"%28-6-x%29%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" >  =   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284-x%29%5E2+ ALIGN=MIDDLE  ALT=\"%284-x%29%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Expand\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=36+%2B+12x+%2B+x%5E2+++=++16+-+8x+%2B+x%5E2 ALIGN=MIDDLE  ALT=\"36+%2B+12x+%2B+x%5E2+++=++16+-+8x+%2B+x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=36+%2B+12x++++=++16+-+8x+ ALIGN=MIDDLE  ALT=\"36+%2B+12x++++=++16+-+8x+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Add 8x to both sides and subtract 36 from both sides\r<BR>\n" );
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document.write( "36 + 12x   =  16 - 8x<BR>\n" );
document.write( "-36  +8x      -36  + 8x\r<BR>\n" );
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document.write( "20x  = - 20\r<BR>\n" );
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document.write( "Divide both sides by 20\r<BR>\n" );
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document.write( " x = -1\r<BR>\n" );
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document.write( "So the x-coordinate is -1\r<BR>\n" );
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document.write( "Using the formula for length of BC  and plugging in the x-coordinate -1\r<BR>\n" );
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document.write( "Distance BC = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%28-6-x%29%5E2+%2B+y%5E2%29 ALIGN=MIDDLE  ALT=\"sqrt%28%28-6-x%29%5E2+%2B+y%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 10\r<BR>\n" );
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document.write( "              <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%28-6-%28-1%29%29%5E2+%2B+y%5E2%29 ALIGN=MIDDLE  ALT=\"sqrt%28%28-6-%28-1%29%29%5E2+%2B+y%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  = 10\r<BR>\n" );
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document.write( "              <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%28-6+%2B+1%29%5E2+%2B+y%5E2%29 ALIGN=MIDDLE  ALT=\"sqrt%28%28-6+%2B+1%29%5E2+%2B+y%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  = 10<BR>\n" );
document.write( "              <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%28-5%29%5E2+%2B+y%5E2%29 ALIGN=MIDDLE  ALT=\"sqrt%28%28-5%29%5E2+%2B+y%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  = 10<BR>\n" );
document.write( "              <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2825+%2B+y%5E2%29 ALIGN=MIDDLE  ALT=\"sqrt%2825+%2B+y%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  = 10<BR>\n" );
document.write( "Square both sides<BR>\n" );
document.write( "              <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2825+%2B+y%5E2%29 ALIGN=MIDDLE  ALT=\"%2825+%2B+y%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  = 100\r<BR>\n" );
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document.write( " Subtract 25 from both sides<BR>\n" );
document.write( "                          <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2 ALIGN=MIDDLE  ALT=\"y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >  = 75<BR>\n" );
document.write( "  Take square root of both sides\r<BR>\n" );
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document.write( "                y = +<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2875%29 ALIGN=MIDDLE  ALT=\"sqrt%2875%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  or y = -<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2875%29 ALIGN=MIDDLE  ALT=\"sqrt%2875%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " y = +<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5sqrt%283%29 ALIGN=MIDDLE  ALT=\"5sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >   or  y = -<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5sqrt%283%29 ALIGN=MIDDLE  ALT=\"5sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The possible coordinates of the third vertex are (-1, +5<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%283%29 ALIGN=MIDDLE  ALT=\"sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >) and (-1, -5<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%283%29 ALIGN=MIDDLE  ALT=\"sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "             \r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );