document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=104648>Question 104648</A>: <I><SPAN STYLE=\"word-break: break-all;\"> (-6,-4),(3,4)  FIND THE DISTANCE BETWEEN EACH PAIR OF POINTS AND THE MIDPOINT OF THE LINE SEGMENT JOINING THE POINTS. LEAVE DISTANCE IN RADICAL FORM.  THANKS </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #76157 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=76157\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \"FIND THE DISTANCE BETWEEN EACH PAIR OF POINTS\"\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(-6,-4\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(3,4\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%28-6-3%29%5E2%2B%28-4-4%29%5E2%29 ALIGN=MIDDLE  ALT=\"d=sqrt%28%28-6-3%29%5E2%2B%28-4-4%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=-6 ALIGN=MIDDLE  ALT=\"x%5B1%5D=-6\"   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=-4 ALIGN=MIDDLE  ALT=\"y%5B1%5D=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B2%5D=4 ALIGN=MIDDLE  ALT=\"y%5B2%5D=4\"   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%28-9%29%5E2%2B%28-8%29%5E2%29 ALIGN=MIDDLE  ALT=\"d=sqrt%28%28-9%29%5E2%2B%28-8%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Evaluate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6-3 ALIGN=MIDDLE  ALT=\"-6-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get -9. Evaluate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4-4 ALIGN=MIDDLE  ALT=\"-4-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get -8. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=sqrt%2881%2B64%29 ALIGN=MIDDLE  ALT=\"d=sqrt%2881%2B64%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%28145%29 ALIGN=MIDDLE  ALT=\"d=sqrt%28145%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=12.0415945787923 ALIGN=MIDDLE  ALT=\"d=12.0415945787923\"   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( "12.04\r<BR>\n" );
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document.write( "So the distance between (-6,-4) and (3,4) is approximately 12.04 units\r<BR>\n" );
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document.write( "\"FIND THE MIDPOINT OF THE LINE SEGMENT JOINING THE POINTS\"\r<BR>\n" );
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document.write( "In order to find the midpoint between the points (-6,-4) and (3,4), 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=%28-6%2B3%29%2F2=%28-3%29%2F2=-1.5 ALIGN=MIDDLE  ALT=\"x%5Bmid%5D=%28-6%2B3%29%2F2=%28-3%29%2F2=-1.5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the x-coordinate of the midpoint is -1.5 (i.e. x=-1.5)\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=%28-4%2B4%29%2F2=%280%29%2F2=0 ALIGN=MIDDLE  ALT=\"y%5Bmid%5D=%28-4%2B4%29%2F2=%280%29%2F2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the y-coordinate of the midpoint is 0 (i.e. y=0)\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=-1.5, y=0, this means the midpoint is (-1.5,0)\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+-8%2C+5%2C+-6%2C+6%2C%0D%0Agraph%28500%2C+500%2C+-8%2C+5%2C+-6%2C+6%2C0%29%2C%0D%0Ablue%28line%28-6%2C-4%2C3%2C4%29%29%2C%0D%0Acircle%28-6%2C-4%2C0.05%29%2C%0D%0Acircle%28-6%2C-4%2C0.08%29%2C%0D%0Acircle%283%2C4%2C0.05%29%2C%0D%0Acircle%283%2C4%2C0.08%29%2C%0D%0Acircle%28-1.5%2C0%2C0.05%29%2C%0D%0Acircle%28-1.5%2C0%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 (-6,-4) and (3,4) with the midpoint (-1.5,0)\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+-8%2C+5%2C+-6%2C+6%2C%0D%0Agraph%28500%2C+500%2C+-8%2C+5%2C+-6%2C+6%2C0%29%2C%0D%0Ablue%28line%28-6%2C-4%2C3%2C4%29%29%2C%0D%0A%0D%0Aline%28-6%2C-4%2C-6%2C%28-4%2B4%29%2F2%29%2C%0D%0Aline%28-6%2C%28-4%2B4%29%2F2%2C-1.5%2C0%29%2C%0D%0Aline%28-1.5%2C0%2C%28-6%2B3%29%2F2%2C4%29%2C%0D%0Aline%28%28-6%2B3%29%2F2%2C4%2C3%2C4%29%2C%0D%0A%0D%0Acircle%28-6%2C-4%2C0.05%29%2C%0D%0Acircle%28-6%2C-4%2C0.08%29%2C%0D%0Acircle%283%2C4%2C0.05%29%2C%0D%0Acircle%283%2C4%2C0.08%29%2C%0D%0Acircle%28-1.5%2C0%2C0.05%29%2C%0D%0Acircle%28-1.5%2C0%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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