document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=93771>Question 93771</A>: <I><SPAN STYLE=\"word-break: break-all;\"> will you please explain to me how to do this using an equation. i am so lost.<BR>\n" );
document.write( " It says, find the distance and the midpoint between the two points: (-2,5) and (3,-1). <BR>\n" );
document.write( "If you could help me, that would be wonderful. thank you and God bless.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #68306 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=68306\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \"find the distance between the two points: (-2,5) and (3,-1).\"\r<BR>\n" );
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document.write( "Start with the given distance formula\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=sqrt%28%28x%5B1%5D-x%5B2%5D%29%5E2%2B%28y%5B1%5D-y%5B2%5D%29%5E2%29 ALIGN=MIDDLE  ALT=\"d=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;\" > where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(x_{1}, y_{1}\right)\"> is the first point <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(-2,5\right)\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(x_{2}, y_{2}\right)\"> is the second point <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(3,-1\right)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=sqrt%28%28-2-3%29%5E2%2B%285--1%29%5E2%29 ALIGN=MIDDLE  ALT=\"d=sqrt%28%28-2-3%29%5E2%2B%285--1%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D=-2 ALIGN=MIDDLE  ALT=\"x%5B1%5D=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D=3 ALIGN=MIDDLE  ALT=\"x%5B2%5D=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B1%5D=5 ALIGN=MIDDLE  ALT=\"y%5B1%5D=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B2%5D=-1 ALIGN=MIDDLE  ALT=\"y%5B2%5D=-1\"   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%28-5%29%5E2%2B%286%29%5E2%29 ALIGN=MIDDLE  ALT=\"d=sqrt%28%28-5%29%5E2%2B%286%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Evaluate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2-3 ALIGN=MIDDLE  ALT=\"-2-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get -5. Evaluate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5--1 ALIGN=MIDDLE  ALT=\"5--1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get 6. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=sqrt%2825%2B36%29 ALIGN=MIDDLE  ALT=\"d=sqrt%2825%2B36%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square each value\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=sqrt%2861%29 ALIGN=MIDDLE  ALT=\"d=sqrt%2861%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add\r<BR>\n" );
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document.write( "So the distance approximates to\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=7.81024967590665 ALIGN=MIDDLE  ALT=\"d=7.81024967590665\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "which rounds to\r<BR>\n" );
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document.write( "7.81\r<BR>\n" );
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document.write( "So the distance between (-2,5) and (3,-1) is approximately 7.81 units\r<BR>\n" );
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document.write( "\"find the midpoint between the two points: (-2,5) and (3,-1)\"\r<BR>\n" );
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document.write( "In order to find the midpoint, we need to average each corresponding coordinate. In other words, we need to add up the corresponding coordinates and divide the sum by 2.\r<BR>\n" );
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document.write( "So lets find the averages between the two points\r<BR>\n" );
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document.write( "To find <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large  \textrm{x_{mid}}\">, average the x-coordinates between the two points\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5Bmid%5D=%28-2%2B3%29%2F2=%281%29%2F2=0.5 ALIGN=MIDDLE  ALT=\"x%5Bmid%5D=%28-2%2B3%29%2F2=%281%29%2F2=0.5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the x-coordinate of the midpoint is 0.5 (i.e. x=0.5)\r<BR>\n" );
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document.write( "To find <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large  \textrm{y_{mid}}\">, average the y-coordinates between the two points\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5Bmid%5D=%285%2B-1%29%2F2=%284%29%2F2=2 ALIGN=MIDDLE  ALT=\"y%5Bmid%5D=%285%2B-1%29%2F2=%284%29%2F2=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the y-coordinate of the midpoint is 2 (i.e. y=2)\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "Since the coordinates of the midpoint are x=0.5, y=2, this means the midpoint is (0.5,2)\r<BR>\n" );
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document.write( "Check:\r<BR>\n" );
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document.write( "Here is a graph to visually see the answer\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28500%2C+500%2C+-4%2C+5%2C+-3%2C+7%2C%0D%0Agraph%28500%2C+500%2C+-4%2C+5%2C+-3%2C+7%2C0%29%2C%0D%0Ablue%28line%28-2%2C5%2C3%2C-1%29%29%2C%0D%0Acircle%28-2%2C5%2C0.05%29%2C%0D%0Acircle%28-2%2C5%2C0.08%29%2C%0D%0Acircle%283%2C-1%2C0.05%29%2C%0D%0Acircle%283%2C-1%2C0.08%29%2C%0D%0Acircle%280.5%2C2%2C0.05%29%2C%0D%0Acircle%280.5%2C2%2C0.08%29%0D%0A%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Graph of the line segment with the endpoints (-2,5) and (3,-1) with the midpoint (0.5,2)\r<BR>\n" );
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document.write( "We could visually verify our answer if we simply draw right triangles from each point like this:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28500%2C+500%2C+-4%2C+5%2C+-3%2C+7%2C%0D%0Agraph%28500%2C+500%2C+-4%2C+5%2C+-3%2C+7%2C0%29%2C%0D%0Ablue%28line%28-2%2C5%2C3%2C-1%29%29%2C%0D%0A%0D%0Aline%28-2%2C5%2C-2%2C%285%2B-1%29%2F2%29%2C%0D%0Aline%28-2%2C%285%2B-1%29%2F2%2C0.5%2C2%29%2C%0D%0Aline%280.5%2C2%2C%28-2%2B3%29%2F2%2C-1%29%2C%0D%0Aline%28%28-2%2B3%29%2F2%2C-1%2C3%2C-1%29%2C%0D%0A%0D%0Acircle%28-2%2C5%2C0.05%29%2C%0D%0Acircle%28-2%2C5%2C0.08%29%2C%0D%0Acircle%283%2C-1%2C0.05%29%2C%0D%0Acircle%283%2C-1%2C0.08%29%2C%0D%0Acircle%280.5%2C2%2C0.05%29%2C%0D%0Acircle%280.5%2C2%2C0.08%29%0D%0A%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Here we can see that the two triangles are congruent, so our answer is verified.\r<BR>\n" );
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