document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=98441>Question 98441</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How would you calculate the following problem????\r<BR>\n" );
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document.write( "Find the length  of AB there is aline above it and the midpoint of ab\r<BR>\n" );
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document.write( "1)  A (-10,6)     2)    A (8,3)<BR>\n" );
document.write( "    B (4,2)             B (7,-2)\r<BR>\n" );
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document.write( "Thanks,\r<BR>\n" );
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document.write( "Amber  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #71606 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=71606\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 1)\r<BR>\n" );
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document.write( "\"Find the length  of AB\"\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(-10,6\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(4,2\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%28-10-4%29%5E2%2B%286-2%29%5E2%29 ALIGN=MIDDLE  ALT=\"d=sqrt%28%28-10-4%29%5E2%2B%286-2%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=-10 ALIGN=MIDDLE  ALT=\"x%5B1%5D=-10\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D=4 ALIGN=MIDDLE  ALT=\"x%5B2%5D=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B1%5D=6 ALIGN=MIDDLE  ALT=\"y%5B1%5D=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B2%5D=2 ALIGN=MIDDLE  ALT=\"y%5B2%5D=2\"   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%28-14%29%5E2%2B%284%29%5E2%29 ALIGN=MIDDLE  ALT=\"d=sqrt%28%28-14%29%5E2%2B%284%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Evaluate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-10-4 ALIGN=MIDDLE  ALT=\"-10-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get -14. Evaluate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6-2 ALIGN=MIDDLE  ALT=\"6-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get 4. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=sqrt%28196%2B16%29 ALIGN=MIDDLE  ALT=\"d=sqrt%28196%2B16%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%28212%29 ALIGN=MIDDLE  ALT=\"d=sqrt%28212%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=2%2Asqrt%2853%29 ALIGN=MIDDLE  ALT=\"d=2%2Asqrt%2853%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify the square root  (note: If you need help with simplifying the square root, check out this <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Radicals/simplifying-square-roots.solver> solver</a>)\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=14.560219778561 ALIGN=MIDDLE  ALT=\"d=14.560219778561\"   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( "14.56\r<BR>\n" );
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document.write( "So the distance between (-10,6) and (4,2) is approximately 14.56 units\r<BR>\n" );
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document.write( "\"Find the midpoint of ab\"\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=%28-10%2B4%29%2F2=%28-6%29%2F2=-3 ALIGN=MIDDLE  ALT=\"x%5Bmid%5D=%28-10%2B4%29%2F2=%28-6%29%2F2=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the x-coordinate of the midpoint is -3 (i.e. x=-3)\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=%286%2B2%29%2F2=%288%29%2F2=4 ALIGN=MIDDLE  ALT=\"y%5Bmid%5D=%286%2B2%29%2F2=%288%29%2F2=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the y-coordinate of the midpoint is 4 (i.e. y=4)\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=-3, y=4, this means the midpoint is (-3,4)\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+-12%2C+6%2C+0%2C+8%2C%0D%0Agraph%28500%2C+500%2C+-12%2C+6%2C+0%2C+8%2C0%29%2C%0D%0Ablue%28line%28-10%2C6%2C4%2C2%29%29%2C%0D%0Acircle%28-10%2C6%2C0.05%29%2C%0D%0Acircle%28-10%2C6%2C0.08%29%2C%0D%0Acircle%284%2C2%2C0.05%29%2C%0D%0Acircle%284%2C2%2C0.08%29%2C%0D%0Acircle%28-3%2C4%2C0.05%29%2C%0D%0Acircle%28-3%2C4%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 (-10,6) and (4,2) with the midpoint (-3,4)\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+-12%2C+6%2C+0%2C+8%2C%0D%0Agraph%28500%2C+500%2C+-12%2C+6%2C+0%2C+8%2C0%29%2C%0D%0Ablue%28line%28-10%2C6%2C4%2C2%29%29%2C%0D%0A%0D%0Aline%28-10%2C6%2C-10%2C%286%2B2%29%2F2%29%2C%0D%0Aline%28-10%2C%286%2B2%29%2F2%2C-3%2C4%29%2C%0D%0Aline%28-3%2C4%2C%28-10%2B4%29%2F2%2C2%29%2C%0D%0Aline%28%28-10%2B4%29%2F2%2C2%2C4%2C2%29%2C%0D%0A%0D%0Acircle%28-10%2C6%2C0.05%29%2C%0D%0Acircle%28-10%2C6%2C0.08%29%2C%0D%0Acircle%284%2C2%2C0.05%29%2C%0D%0Acircle%284%2C2%2C0.08%29%2C%0D%0Acircle%28-3%2C4%2C0.05%29%2C%0D%0Acircle%28-3%2C4%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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document.write( "2)\r<BR>\n" );
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document.write( "\"Find the length  of AB\"\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(8,3\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(7,-2\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%288-7%29%5E2%2B%283--2%29%5E2%29 ALIGN=MIDDLE  ALT=\"d=sqrt%28%288-7%29%5E2%2B%283--2%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=8 ALIGN=MIDDLE  ALT=\"x%5B1%5D=8\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D=7 ALIGN=MIDDLE  ALT=\"x%5B2%5D=7\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B1%5D=3 ALIGN=MIDDLE  ALT=\"y%5B1%5D=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B2%5D=-2 ALIGN=MIDDLE  ALT=\"y%5B2%5D=-2\"   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%281%29%5E2%2B%285%29%5E2%29 ALIGN=MIDDLE  ALT=\"d=sqrt%28%281%29%5E2%2B%285%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Evaluate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8-7 ALIGN=MIDDLE  ALT=\"8-7\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get 1. Evaluate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3--2 ALIGN=MIDDLE  ALT=\"3--2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get 5. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=sqrt%281%2B25%29 ALIGN=MIDDLE  ALT=\"d=sqrt%281%2B25%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%2826%29 ALIGN=MIDDLE  ALT=\"d=sqrt%2826%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=5.09901951359278 ALIGN=MIDDLE  ALT=\"d=5.09901951359278\"   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( "5.1\r<BR>\n" );
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document.write( "So the distance between (8,3) and (7,-2) is approximately 5.1 units\r<BR>\n" );
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document.write( "\"Find the midpoint of ab\"\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=%288%2B7%29%2F2=%2815%29%2F2=7.5 ALIGN=MIDDLE  ALT=\"x%5Bmid%5D=%288%2B7%29%2F2=%2815%29%2F2=7.5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the x-coordinate of the midpoint is 7.5 (i.e. x=7.5)\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=%283%2B-2%29%2F2=%281%29%2F2=0.5 ALIGN=MIDDLE  ALT=\"y%5Bmid%5D=%283%2B-2%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 y-coordinate of the midpoint is 0.5 (i.e. y=0.5)\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=7.5, y=0.5, this means the midpoint is (7.5,0.5)\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+5%2C+10%2C+-4%2C+5%2C%0D%0Agraph%28500%2C+500%2C+5%2C+10%2C+-4%2C+5%2C0%29%2C%0D%0Ablue%28line%288%2C3%2C7%2C-2%29%29%2C%0D%0Acircle%288%2C3%2C0.05%29%2C%0D%0Acircle%288%2C3%2C0.08%29%2C%0D%0Acircle%287%2C-2%2C0.05%29%2C%0D%0Acircle%287%2C-2%2C0.08%29%2C%0D%0Acircle%287.5%2C0.5%2C0.05%29%2C%0D%0Acircle%287.5%2C0.5%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 (8,3) and (7,-2) with the midpoint (7.5,0.5)\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+5%2C+10%2C+-4%2C+5%2C%0D%0Agraph%28500%2C+500%2C+5%2C+10%2C+-4%2C+5%2C0%29%2C%0D%0Ablue%28line%288%2C3%2C7%2C-2%29%29%2C%0D%0A%0D%0Aline%288%2C3%2C8%2C%283%2B-2%29%2F2%29%2C%0D%0Aline%288%2C%283%2B-2%29%2F2%2C7.5%2C0.5%29%2C%0D%0Aline%287.5%2C0.5%2C%288%2B7%29%2F2%2C-2%29%2C%0D%0Aline%28%288%2B7%29%2F2%2C-2%2C7%2C-2%29%2C%0D%0A%0D%0Acircle%288%2C3%2C0.05%29%2C%0D%0Acircle%288%2C3%2C0.08%29%2C%0D%0Acircle%287%2C-2%2C0.05%29%2C%0D%0Acircle%287%2C-2%2C0.08%29%2C%0D%0Acircle%287.5%2C0.5%2C0.05%29%2C%0D%0Acircle%287.5%2C0.5%2C0.08%29%0D%0A%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Here we can see that the two triangles are congruent, so our answer is verified.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );