document.write( "Question 780900: The midpoints of the sides of a triangle are at (0,-3), (-1/2,7/2) and (-7/2,1/2). Find the coordinates of the vertices \n" ); document.write( "
Algebra.Com's Answer #475784 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
\r\n" );
document.write( "In a message dated 9/5/2013 9:00:27 A.M. Eastern Daylight Time, AnlytcPhil@aol.com writes:\r\n" );
document.write( " (0,-3), (-1/2,7/2) and (-7/2,1/2)\r\n" );
document.write( "\r\n" );
document.write( "Plot the points, and connect them:\r\n" );
document.write( "\r\n" );
document.write( " \r\n" );
document.write( "\r\n" );
document.write( "These green lines are the mid-segments of the triangle we are trying\r\n" );
document.write( "to determine the coordinates of.\r\n" );
document.write( "\r\n" );
document.write( "We remember that a mid-segment of a triangle (which joins the \r\n" );
document.write( "midpoints of two sides of a triangle) is parallel to the third \r\n" );
document.write( "side of the triangle.  The mid-segment is also 1/2 of the third\r\n" );
document.write( "side , but we do not need that fact.\r\n" );
document.write( "\r\n" );
document.write( "Through each of those points we will find the equation of the line \r\n" );
document.write( "paralell to the mid-segment joining the other two points:\r\n" );
document.write( "\r\n" );
document.write( "We find the slope of the mid-segment joining\r\n" );
document.write( "\r\n" );
document.write( "(0,-3) and  (\"-1%2F2\",\"7%2F2\")\r\n" );
document.write( "\r\n" );
document.write( "m = \"%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29\"\r\n" );
document.write( "\r\n" );
document.write( "where (x1,y1) = (0,-3)\r\n" );
document.write( "and where (x2,y2) = (\"-7%2F2\",\"1%2F2\")\r\n" );
document.write( "\r\n" );
document.write( "m = \"%28%287%2F2%29-%28-3%29%29%2F%28%28-1%2F2%29-%280%29%29\" = \"%287%2F2%2B3%29%2F%28-1%2F2-0%29\" = \"%2813%2F2%29%2F%28-1%2F2%29\" = \"13%2F2\"÷\"-1%2F2\" = \"13%2F2\"×\"-2%2F1\" = \"13%2Fcross%282%29\"×\"-cross%282%29%2F1\" = -13\r\n" );
document.write( "\r\n" );
document.write( "Now we find the equation of the line through the third point (\"-7%2F2\",\"1%2F2\"):\r\n" );
document.write( "\r\n" );
document.write( "Point-slope formula:\r\n" );
document.write( "\r\n" );
document.write( "y - y1 = m(x - x1)\r\n" );
document.write( "\r\n" );
document.write( "where (x1,y1) = the third point (\"-7%2F2\",\"1%2F2\")\r\n" );
document.write( "\r\n" );
document.write( "y - \"1%2F2\" = -13(x - (\"-7%2F2\"))\r\n" );
document.write( "\r\n" );
document.write( "y - \"1%2F2\" = -13(x + \"7%2F2\")\r\n" );
document.write( "\r\n" );
document.write( "y - \"1%2F2\" = -13x - \"91%2F2\")\r\n" );
document.write( "\r\n" );
document.write( "    y = -13x - \"90%2F2\"\r\n" );
document.write( "\r\n" );
document.write( "    y = -13x - 45  \r\n" );
document.write( "\r\n" );
document.write( "That line is the red one below:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "-------------------\r\n" );
document.write( "\r\n" );
document.write( "We find the slope of the mid-segment joining\r\n" );
document.write( "\r\n" );
document.write( "(0,-3) and (\"-7%2F2\",\"1%2F2\")\r\n" );
document.write( "\r\n" );
document.write( "m = \"%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29\"\r\n" );
document.write( "\r\n" );
document.write( "where (x1,y1) = (0,-3)\r\n" );
document.write( "and where (x2,y2) = (\"-7%2F2\",\"1%2F2\")\r\n" );
document.write( "\r\n" );
document.write( "m = \"%28%281%2F2%29-%28-3%29%29%2F%28%28-7%2F2%29-%280%29%29\" = \"%281%2F2%2B3%29%2F%28-7%2F2-0%29\" = \"%281%2F2%2B6%2F2%29%2F%28-7%2F2%29\" = \"%287%2F2%29%2F%28-7%2F2%29\" = \"7%2F2\"÷\"-7%2F2\" = -1\r\n" );
document.write( "\r\n" );
document.write( "Now we find the equation of the line through the third point (\"-1%2F2\",\"7%2F2\"):\r\n" );
document.write( "\r\n" );
document.write( "Point-slope formula:\r\n" );
document.write( "\r\n" );
document.write( "y - y1 = m(x - x1)\r\n" );
document.write( "\r\n" );
document.write( "where (x1,y1) = the third point (\"-1%2F2\",\"7%2F2\")\r\n" );
document.write( "\r\n" );
document.write( "y - \"7%2F2\" = -1(x - (\"-1%2F2\"))\r\n" );
document.write( "\r\n" );
document.write( "y - \"7%2F2\" = -1(x + \"1%2F2\")\r\n" );
document.write( "\r\n" );
document.write( "y - \"7%2F2\" = -x - \"1%2F2\")\r\n" );
document.write( "\r\n" );
document.write( "    y = -x + \"6%2F2\"\r\n" );
document.write( "\r\n" );
document.write( "    y = -x + 3  \r\n" );
document.write( "\r\n" );
document.write( " \r\n" );
document.write( "\r\n" );
document.write( "That line is the second red one below:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "-------------------\r\n" );
document.write( "\r\n" );
document.write( "We find the slope of the mid-segment joining\r\n" );
document.write( "\r\n" );
document.write( "(\"-1%2F2\",\"7%2F2\") and (\"-7%2F2\",\"1%2F2\")\r\n" );
document.write( "\r\n" );
document.write( "m = \"%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29\"\r\n" );
document.write( "\r\n" );
document.write( "where (x1,y1) = (0,-3)\r\n" );
document.write( "and where (x2,y2) = (\"-7%2F2\",\"1%2F2\")\r\n" );
document.write( "\r\n" );
document.write( "m = \"%28%281%2F2%29-%287%2F2%29%29%2F%28%28-7%2F2%29-%28-1%2F2%29%29\" = \"%281%2F2-7%2F2%29%2F%28-7%2F2%2B1%2F2%29\" = \"%28-6%2F2%29%2F%28-6%2F2%29\" = \"%28-3%29%2F%28-3%29\" = 1\r\n" );
document.write( "\r\n" );
document.write( "Now we find the equation of the line through the third point  (0,-3), :\r\n" );
document.write( "\r\n" );
document.write( "Point-slope formula:\r\n" );
document.write( "\r\n" );
document.write( "y - y1 = m(x - x1)\r\n" );
document.write( "\r\n" );
document.write( "where (x1,y1) = the third point (\"-1%2F2\",\"7%2F2\")\r\n" );
document.write( "\r\n" );
document.write( "y - (-3)} = 1(x - 0)\r\n" );
document.write( "\r\n" );
document.write( "y + 3 = x \r\n" );
document.write( "\r\n" );
document.write( "    y = x - 3  \r\n" );
document.write( "\r\n" );
document.write( " \r\n" );
document.write( "\r\n" );
document.write( "That line is the second red one below:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now we find the three points of intersection of the three pairs of\r\n" );
document.write( "the lines\r\n" );
document.write( "\r\n" );
document.write( "y = -13x - 45,  y = -x + 3, y = x - 3\r\n" );
document.write( "\r\n" );
document.write( "Solve these three systems:\r\n" );
document.write( "\r\n" );
document.write( "y = -13x - 45      y = -13x - 45      y = -x + 3   \r\n" );
document.write( "y = -x + 3         y = x - 3          y = x - 3\r\n" );
document.write( "\r\n" );
document.write( "You can do that.  They are\r\n" );
document.write( "\r\n" );
document.write( "(-4,7),  (-3,-6), and (3,0)\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" ); document.write( "
\n" );