document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1203326>Question 1203326</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the point y-axis that is equidistant from (5,1) and (-3, -1) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #838790 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3816)</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=838790\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Answer: <font color=red size=4>(0,4)</font>\r<BR>\n" );
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document.write( "Explanation\r<BR>\n" );
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document.write( "I'll go over two methods.\r<BR>\n" );
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document.write( "Method 1\r<BR>\n" );
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document.write( "The given points are (5,1) and (-3,-1) which I'll label A and B respectively.\r<BR>\n" );
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document.write( "The mystery point is of the form (0,y) residing on the y axis.<BR>\n" );
document.write( "Let's call this point C.\r<BR>\n" );
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document.write( "A = (5,1)<BR>\n" );
document.write( "B = (-3,-1)<BR>\n" );
document.write( "C = (0,y)\r<BR>\n" );
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document.write( "The goal is to determine the value of y such that segment AC and segment BC have the same length.\r<BR>\n" );
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document.write( "Compute the distance from A to C<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+sqrt%28+%28x2-x1%29%5E2%2B%28y2-y1%29%5E2+%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%28+%28x2-x1%29%5E2%2B%28y2-y1%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=d+=+sqrt%28+%280-5%29%5E2%2B%28y-1%29%5E2+%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%28+%280-5%29%5E2%2B%28y-1%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=d+=+sqrt%28+25%2B%28y-1%29%5E2+%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%28+25%2B%28y-1%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=d+=+sqrt%28+y%5E2-2y%2B26+%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%28+y%5E2-2y%2B26+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Then do the same for the distance from B to C<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+sqrt%28+%280-%28-3%29%29%5E2%2B%28y-%28-1%29%29%5E2+%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%28+%280-%28-3%29%29%5E2%2B%28y-%28-1%29%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=d+=+sqrt%28+%280%2B3%29%5E2%2B%28y%2B1%29%5E2+%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%28+%280%2B3%29%5E2%2B%28y%2B1%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=d+=+sqrt%28+%283%29%5E2%2B%28y%2B1%29%5E2+%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%28+%283%29%5E2%2B%28y%2B1%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=d+=+sqrt%28+9%2B%28y%5E2%2B2y%2B1%29+%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%28+9%2B%28y%5E2%2B2y%2B1%29+%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=d+=+sqrt%28+y%5E2%2B2y%2B10+%29 ALIGN=MIDDLE  ALT=\"d+=+sqrt%28+y%5E2%2B2y%2B10+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Equate those expressions to allow us to solve for y.\r<BR>\n" );
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document.write( "distance A to C = distance B to C\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28+y%5E2-2y%2B26+%29+=+sqrt%28+y%5E2%2B2y%2B10+%29 ALIGN=MIDDLE  ALT=\"sqrt%28+y%5E2-2y%2B26+%29+=+sqrt%28+y%5E2%2B2y%2B10+%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=y%5E2-2y%2B26+=+y%5E2%2B2y%2B10 ALIGN=MIDDLE  ALT=\"y%5E2-2y%2B26+=+y%5E2%2B2y%2B10\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2y%2B26+=+2y%2B10 ALIGN=MIDDLE  ALT=\"-2y%2B26+=+2y%2B10\"   DISABLED_style= \"max-width:90%; height:auto;\" > The y^2 terms cancel\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2y-2y+=+10-26 ALIGN=MIDDLE  ALT=\"-2y-2y+=+10-26\"   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=-4y+=+-16 ALIGN=MIDDLE  ALT=\"-4y+=+-16\"   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=y+=+-16%2F%28-4%29 ALIGN=MIDDLE  ALT=\"y+=+-16%2F%28-4%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=y+=+4 ALIGN=MIDDLE  ALT=\"y+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "We go from the template (0,y) to (0,4)<BR>\n" );
document.write( "The point <font color=red size=4>(0,4)</font> is equidistant from A(5,1) and B(-3,-1) such that it is on the y axis.\r<BR>\n" );
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document.write( "Method 2\r<BR>\n" );
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document.write( "The given points are<BR>\n" );
document.write( "A = (5,1)<BR>\n" );
document.write( "B = (-3,-1)\r<BR>\n" );
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document.write( "Compute the midpoint of segment AB.<BR>\n" );
document.write( "We add up the coordinates and divide in half<BR>\n" );
document.write( "x: (5+(-3))/2 = 1<BR>\n" );
document.write( "y: (1+(-1))/2 = 0\r<BR>\n" );
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document.write( "The midpoint of segment AB is located at (1,0) which I'll call point D.\r<BR>\n" );
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document.write( "Now compute the slope of line AB<BR>\n" );
document.write( "m = (y2-y1)/(x2-x1)<BR>\n" );
document.write( "m = (-1-1)/(-3-5)<BR>\n" );
document.write( "m = (-2)/(-8)<BR>\n" );
document.write( "m = 1/4<BR>\n" );
document.write( "Line AB has a slope of 1/4. <BR>\n" );
document.write( "The perpendicular line has a slope of -4/1 = -4. Apply the negative reciprocal. We flip the fraction and flip the sign.\r<BR>\n" );
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document.write( "The goal now is to find the equation of the perpendicular line through the midpoint D(1,0)\r<BR>\n" );
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document.write( "In other words: we want the equation of the perpendicular bisector of segment AB.\r<BR>\n" );
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document.write( "Apply point-slope form<BR>\n" );
document.write( "y - y1 = m(x - x1)<BR>\n" );
document.write( "y - 0 = -4(x - 1)<BR>\n" );
document.write( "y = -4x + 4<BR>\n" );
document.write( "This equation has slope -4 and y-intercept 4\r<BR>\n" );
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document.write( "The y intercept of this equation is the exact point equidistant from A and B, and is on the y axis. <BR>\n" );
document.write( "I'll leave the proof to the reader as to why this works. A hint would be to draw segments CA, CB, and CD. <BR>\n" );
document.write( "Then prove triangle ACD is congruent to triangle BCD.\r<BR>\n" );
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document.write( "I recommend using graphing tools like GeoGebra or Desmos. <BR>\n" );
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