document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1208377>Question 1208377</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the equation of the circle passing through the points (1,2),(3,6),(5,4).\r<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 #846769 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=846769\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Answer: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-8%2F3%29%5E2%2B%28y-11%2F3%29%5E2=50%2F9 ALIGN=MIDDLE  ALT=\"%28x-8%2F3%29%5E2%2B%28y-11%2F3%29%5E2=50%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Explanation\r<BR>\n" );
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document.write( "Let<BR>\n" );
document.write( "A = (1,2)<BR>\n" );
document.write( "B = (3,6)<BR>\n" );
document.write( "C = (5,4)<BR>\n" );
document.write( "The order of the points does not matter. <BR>\n" );
document.write( "The labels are there to help set up the other items mentioned in this solution page.\r<BR>\n" );
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document.write( "The circle you're trying to find is known as a <a  rel=nofollow HREF=\"https://www.mathopenref.com/trianglecircumcircle.html\">circumcircle</a> aka circumscribed circle.<BR>\n" );
document.write( "The circumcenter is the center of the circumcircle.<BR>\n" );
document.write( "The circumcenter is at the intersection of the perpendicular bisectors.<BR>\n" );
document.write( "Check out <a  rel=nofollow HREF=\"https://www.algebra.com/algebra/homework/Linear-equations/perpendicular-bisector-example1.lesson\">this page</a> to see how to find the equation of the perpendicular bisector of AB.<BR>\n" );
document.write( "That page concludes with the equations x+2y=10 and y=(-1/2)x+5.<BR>\n" );
document.write( "The equation I'll use is x+2y=10.\r<BR>\n" );
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document.write( "Follow a similar process to determine an equation for the perpendicular bisector of BC is x-y=-1.\r<BR>\n" );
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document.write( "Solving this system <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x%2B2y+=+10%2Cx-y+=+-1%29 ALIGN=MIDDLE  ALT=\"system%28x%2B2y+=+10%2Cx-y+=+-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > will pinpoint the circumcenter. \r<BR>\n" );
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document.write( "Subtract the equations. <BR>\n" );
document.write( "This is to cancel the x terms so you can solve for y.<BR>\n" );
document.write( "You should get 3y = 11 which solves to y = 11/3. <BR>\n" );
document.write( "This is the y coordinate of the center of the circle.<BR>\n" );
document.write( "Plug y = 11/3 into either x+2y=10 or x-y=-1. Solve for x. You should get x = 8/3.<BR>\n" );
document.write( "I'll leave the scratch work for the student to do. \r<BR>\n" );
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document.write( "The center of the circle is located at (h,k) = (8/3,11/3)<BR>\n" );
document.write( "h = 8/3 = 2.6667 approximately<BR>\n" );
document.write( "k = 11/3 = 3.6667 approximately<BR>\n" );
document.write( "For each decimal the '6's go on forever but we have to round at some point.\r<BR>\n" );
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document.write( "So we have the center of this circle. <BR>\n" );
document.write( "We now need the radius.\r<BR>\n" );
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document.write( "Use the distance formula<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+sqrt%28+%28x%5B2%5D-x%5B1%5D%29%5E2%2B%28y%5B2%5D-y%5B1%5D%29%5E2+%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%28+%28x%5B2%5D-x%5B1%5D%29%5E2%2B%28y%5B2%5D-y%5B1%5D%29%5E2+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "to calculate the distance from the center (8/3,11/3) to any of the three points A,B or C.<BR>\n" );
document.write( "This will yield the radius r. <BR>\n" );
document.write( "I'll let the student do the scratch work.<BR>\n" );
document.write( "You should get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+%285%2F3%29%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"r+=+%285%2F3%29%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Square both sides to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5E2+=+50%2F9 ALIGN=MIDDLE  ALT=\"r%5E2+=+50%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Side note: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285%2F3%29%2Asqrt%282%29=2.357023 ALIGN=MIDDLE  ALT=\"%285%2F3%29%2Asqrt%282%29=2.357023\"   DISABLED_style= \"max-width:90%; height:auto;\" > approximately and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=50%2F9+=+5.55556 ALIGN=MIDDLE  ALT=\"50%2F9+=+5.55556\"   DISABLED_style= \"max-width:90%; height:auto;\" > approximately<BR>\n" );
document.write( "The decimal 5.55556 should have the '5's go on forever, but we have to round at some point.\r<BR>\n" );
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document.write( "Summary:\r<BR>\n" );
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document.write( "(h,k) = (8/3, 11/3) is the center<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+%285%2F3%29%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"r+=+%285%2F3%29%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the radius which has both sides square to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5E2+=+50%2F9 ALIGN=MIDDLE  ALT=\"r%5E2+=+50%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The circle template <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-h%29%5E2%2B%28y-k%29%5E2=r%5E2 ALIGN=MIDDLE  ALT=\"%28x-h%29%5E2%2B%28y-k%29%5E2=r%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > updates to the final answer <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-8%2F3%29%5E2%2B%28y-11%2F3%29%5E2=50%2F9 ALIGN=MIDDLE  ALT=\"%28x-8%2F3%29%5E2%2B%28y-11%2F3%29%5E2=50%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "You can verify by using <a  rel=nofollow HREF=\"https://www.geogebra.org/calculator\">GeoGebra</a> or <a  rel=nofollow HREF=\"https://www.desmos.com/calculator\">Desmos</a> <BR>\n" );
document.write( "There are many other similar tools.\r<BR>\n" );
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document.write( "Another way to verify would be to plug the coordinates of points A,B,C into the answer equation. Simplifying everything should lead to the same number on each side.<BR>\n" );
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