document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=771666>Question 771666</A>: <I><SPAN STYLE=\"word-break: break-all;\"> find the center and radius of the circle passes through (2,3), (6,1) and (4,3). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #470487 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=470487\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The center of the circle is at the same distance from the 3 points.<BR>\n" );
document.write( "That means it's on the perpendicular bisector of the segments connecting those 3 points.<BR>\n" );
document.write( "Points (2,3) and (4,3) are on a horizontal segment with midpoint at (3,3).<BR>\n" );
document.write( "The perpendicular bisector of that segment is the vertical line that passes through (3,3), with equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=3 ALIGN=MIDDLE  ALT=\"x=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "All we need now is a second segment and its perpendicular bisector.<BR>\n" );
document.write( "The midpoint of the segment connecting (6,1) and (4,3) is<BR>\n" );
document.write( "(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%286%2B4%29%2F2 ALIGN=MIDDLE  ALT=\"%286%2B4%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2B3%29%2F2 ALIGN=MIDDLE  ALT=\"%281%2B3%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >) or (5,2).<BR>\n" );
document.write( "The slope of the line connecting (6,1) and (4,3) is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283-1%29%2F%284-6%29=2%2F%28-2%29=-1 ALIGN=MIDDLE  ALT=\"%283-1%29%2F%284-6%29=2%2F%28-2%29=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The slopes of perpendicular line multiply to yield <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so the slope of the perpendicular bisector to the segment connecting (6,1) and (4,3) is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-1%29%2F%28-1%29=1 ALIGN=MIDDLE  ALT=\"%28-1%29%2F%28-1%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The perpendicular bisector to the segment connecting (6,1) and (4,3) has the equation<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-2=1%28x-5%29 ALIGN=MIDDLE  ALT=\"y-2=1%28x-5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in point-slope dorm based on midpoint (5,2).<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-2=1%28x-5%29 ALIGN=MIDDLE  ALT=\"y-2=1%28x-5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-2=x-5 ALIGN=MIDDLE  ALT=\"y-2=x-5\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x-3 ALIGN=MIDDLE  ALT=\"y=x-3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The intersection of the two perpendicular bisectors found is the solution to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=3%2Cy=x-3%29 ALIGN=MIDDLE  ALT=\"system%28x=3%2Cy=x-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and that is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=3%2Cy=0%29 ALIGN=MIDDLE  ALT=\"system%28x=3%2Cy=0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the point (3,0),<BR>\n" );
document.write( "The center of the circle is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28%22%283%2C0%29%22%29 ALIGN=MIDDLE  ALT=\"highlight%28%22%283%2C0%29%22%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "The radius is the distance from that point to any of the 3 given points.<BR>\n" );
document.write( "For example, using (4,3),<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=radius=sqrt%28%284-3%29%5E2%2B%283-0%29%5E2%29 ALIGN=MIDDLE  ALT=\"radius=sqrt%28%284-3%29%5E2%2B%283-0%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=radius=sqrt%281%5E2%2B3%5E2%29 ALIGN=MIDDLE  ALT=\"radius=sqrt%281%5E2%2B3%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=radius=sqrt%281%2B9%29 ALIGN=MIDDLE  ALT=\"radius=sqrt%281%2B9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28radius=sqrt%2810%29%29 ALIGN=MIDDLE  ALT=\"highlight%28radius=sqrt%2810%29%29\"   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%2C-5%2C5%2C%0D%0Agrid%281%29%2Cred%28circle%283%2C0%2C0.1%29%29%2C%0D%0Ablue%28circle%283%2C0%2Csqrt%2810%29%29%29%2C%0D%0Ablue%28circle%282%2C3%2C0.1%29%29%2Cblue%28circle%284%2C3%2C0.1%29%29%2Cblue%28circle%286%2C1%2C0.1%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );