document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=149703>Question 149703</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find coordinates for the centroid of the triangle whose vertices are (a) (-1,5), (-2,8) and (3,3); b) (2,7),(8,1) and (14,11); (c) (a,p), (b,q) and (c,r). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #109814 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=109814\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "The formula for the coordinates of the centroid is \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%281%2F3%29%28x%5Ba%5D%2Bx%5Bb%5D%2Bx%5Bc%5D%29 ALIGN=MIDDLE  ALT=\"x=%281%2F3%29%28x%5Ba%5D%2Bx%5Bb%5D%2Bx%5Bc%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%2F3%29%28y%5Ba%5D%2By%5Bb%5D%2By%5Bc%5D%29 ALIGN=MIDDLE  ALT=\"y=%281%2F3%29%28y%5Ba%5D%2By%5Bb%5D%2By%5Bc%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "note: notice how we're simply averaging the coordinates\r<BR>\n" );
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document.write( "where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(x_{a},y_{a}\right) \"> ,  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(x_{b},y_{b}\right) \">, and  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(x_{c},y_{c}\right) \">  are the coordinates of the three vertices a, b, and c respectively.\r<BR>\n" );
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document.write( "a)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%281%2F3%29%28x%5Ba%5D%2Bx%5Bb%5D%2Bx%5Bc%5D%29 ALIGN=MIDDLE  ALT=\"x=%281%2F3%29%28x%5Ba%5D%2Bx%5Bb%5D%2Bx%5Bc%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the formula for finding the x-coordinate of the centroid.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%281%2F3%29%28-1-2%2B3%29 ALIGN=MIDDLE  ALT=\"x=%281%2F3%29%28-1-2%2B3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in the x-coordinates of the given points\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%281%2F3%29%280%29 ALIGN=MIDDLE  ALT=\"x=%281%2F3%29%280%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=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply.\r<BR>\n" );
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document.write( "So the x-coordinate of the centroid is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "------\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%2F3%29%28y%5Ba%5D%2By%5Bb%5D%2By%5Bc%5D%29 ALIGN=MIDDLE  ALT=\"y=%281%2F3%29%28y%5Ba%5D%2By%5Bb%5D%2By%5Bc%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the formula for finding the y-coordinate of the centroid.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%2F3%29%285%2B8%2B3%29 ALIGN=MIDDLE  ALT=\"y=%281%2F3%29%285%2B8%2B3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in the x-coordinates of the given points\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%2F3%29%2816%29 ALIGN=MIDDLE  ALT=\"y=%281%2F3%29%2816%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=y=5.33 ALIGN=MIDDLE  ALT=\"y=5.33\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify.\r<BR>\n" );
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document.write( "So the y-coordinate of the centroid is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=5.33 ALIGN=MIDDLE  ALT=\"y=5.33\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the centroid is (0,5.33)\r<BR>\n" );
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document.write( "b)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%281%2F3%29%28x%5Ba%5D%2Bx%5Bb%5D%2Bx%5Bc%5D%29 ALIGN=MIDDLE  ALT=\"x=%281%2F3%29%28x%5Ba%5D%2Bx%5Bb%5D%2Bx%5Bc%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the formula for finding the x-coordinate of the centroid.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%281%2F3%29%282%2B8%2B14%29 ALIGN=MIDDLE  ALT=\"x=%281%2F3%29%282%2B8%2B14%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in the x-coordinates of the given points\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%281%2F3%29%2824%29 ALIGN=MIDDLE  ALT=\"x=%281%2F3%29%2824%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=x=8 ALIGN=MIDDLE  ALT=\"x=8\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply.\r<BR>\n" );
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document.write( "So the x-coordinate of the centroid is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=8 ALIGN=MIDDLE  ALT=\"x=8\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "------\r<BR>\n" );
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document.write( "(2,7),(8,1) and (14,11)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%2F3%29%28y%5Ba%5D%2By%5Bb%5D%2By%5Bc%5D%29 ALIGN=MIDDLE  ALT=\"y=%281%2F3%29%28y%5Ba%5D%2By%5Bb%5D%2By%5Bc%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the formula for finding the y-coordinate of the centroid.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%2F3%29%287%2B1%2B11%29 ALIGN=MIDDLE  ALT=\"y=%281%2F3%29%287%2B1%2B11%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in the x-coordinates of the given points\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%2F3%29%2819%29 ALIGN=MIDDLE  ALT=\"y=%281%2F3%29%2819%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=y=6.33 ALIGN=MIDDLE  ALT=\"y=6.33\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify.\r<BR>\n" );
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document.write( "So the y-coordinate of the centroid is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=6.33 ALIGN=MIDDLE  ALT=\"y=6.33\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the centroid is (0,5.33)\r<BR>\n" );
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document.write( "c)\r<BR>\n" );
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document.write( "(a,p), (b,q) and (c,r).\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%281%2F3%29%28x%5Ba%5D%2Bx%5Bb%5D%2Bx%5Bc%5D%29 ALIGN=MIDDLE  ALT=\"x=%281%2F3%29%28x%5Ba%5D%2Bx%5Bb%5D%2Bx%5Bc%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the formula for finding the x-coordinate of the centroid.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%281%2F3%29%28a%2Bb%2Bc%29 ALIGN=MIDDLE  ALT=\"x=%281%2F3%29%28a%2Bb%2Bc%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in the x-coordinates of the given points\r<BR>\n" );
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document.write( "So the x-coordinate of the centroid is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%281%2F3%29%28a%2Bb%2Bc%29 ALIGN=MIDDLE  ALT=\"x=%281%2F3%29%28a%2Bb%2Bc%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "------\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%281%2F3%29%28x%5Ba%5D%2Bx%5Bb%5D%2Bx%5Bc%5D%29 ALIGN=MIDDLE  ALT=\"x=%281%2F3%29%28x%5Ba%5D%2Bx%5Bb%5D%2Bx%5Bc%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the formula for finding the x-coordinate of the centroid.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%281%2F3%29%28p%2Bq%2Br%29 ALIGN=MIDDLE  ALT=\"x=%281%2F3%29%28p%2Bq%2Br%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in the x-coordinates of the given points\r<BR>\n" );
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document.write( "So the y-coordinate of the centroid is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%2F3%29%28p%2Bq%2Br%29 ALIGN=MIDDLE  ALT=\"y=%281%2F3%29%28p%2Bq%2Br%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the centroid is (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F3%29%28a%2Bb%2Bc%29 ALIGN=MIDDLE  ALT=\"%281%2F3%29%28a%2Bb%2Bc%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F3%29%28p%2Bq%2Br%29 ALIGN=MIDDLE  ALT=\"%281%2F3%29%28p%2Bq%2Br%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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