document.write( "Question 1084770: Triangle ABC has vertices A(-2, -3), B(2, 5), and C(6, -7). Find the circumcentre and the centroid. \n" ); document.write( "
Algebra.Com's Answer #698962 by Edwin McCravy(20059) $\"\"$ $\"About$
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document.write( "Let the equation of the circle be\r\n" );
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document.write( "Substituting each of the points, we have this system\r\n" );
document.write( "of three equations:\r\n" );
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document.write( "Subtracting the first two equations\r\n" );
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document.write( "Factoring both sides as the difference of squares:\r\n" );
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document.write( "Divide both sides by 8\r\n" );
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document.write( "  <--equation A \r\n" );
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document.write( "Subtracting the second and third equations\r\n" );
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document.write( "Factoring both sides as the difference of squares:\r\n" );
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document.write( "Divide both sides by -4\r\n" );
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document.write( "Divide both sides by 2\r\n" );
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document.write( "  <--equation B\r\n" );
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document.write( "Setting the expressions for h equal in equations A and B:\r\n" );
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document.write( "Substituting in equation B\r\n" );
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document.write( "So the circumcenter is (h,k) = (4,-1)\r\n" );
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document.write( "The centroid is the point where the three medians intersect.\r\n" );
document.write( "So we will find the equations of two medians and then solve \r\n" );
document.write( "the system of equations to find their point of intersection\r\n" );
document.write( "which will be the centroid.\r\n" );
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document.write( "First we find the midpoint of AB\r\n" );
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document.write( " <--midpoint of AB\r\n" );
document.write( "We will label that midpoint D\r\n" );
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document.write( "Next we draw the median CD from the vertex C(6,-7)\r\n" );
document.write( "to the midpoint D(0,1) of the opposite side AB:\r\n" );
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document.write( "Next we find the equation of that line. \r\n" );
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document.write( "So we find the slope (gradient) of median CD\r\n" );
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document.write( "  <-- slope of CD\r\n" );
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document.write( "Then we use the point-slope formula:\r\n" );
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document.write( "  <-- equation of CD \r\n" );
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document.write( "---------------------\r\n" );
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document.write( "Now we find the equation of one of the other two medians,\r\n" );
document.write( "say the median from A to BC \r\n" );
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document.write( "We find the midpoint of BC\r\n" );
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document.write( " <--midpoint of BC\r\n" );
document.write( "We will label that midpoint E(4,-1).\r\n" );
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document.write( "[Notice that E(4,-1) just happens to be the circumcenter, the center\r\n" );
document.write( "of the circle in the first part, which means that BC was a diameter\r\n" );
document.write( "of the circumscribing circle in the first part.  But that is \r\n" );
document.write( "irrelevant to this part, just interesting.]\r\n" );
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document.write( "So we draw the median AE from the vertex A(-2,-3)\r\n" );
document.write( "to the midpoint E(4,-1) of the opposite side AB:\r\n" );
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document.write( "Next we find the equation of the median AE. \r\n" );
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document.write( "So we find the slope (gradient) of median AE\r\n" );
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document.write( "  <-- slope of median AE\r\n" );
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document.write( "Then we use the point-slope formula:\r\n" );
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document.write( "  <-- equation of AE \r\n" );
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document.write( "To find the centroid we solve this system of\r\n" );
document.write( "equations to find the point where those two\r\n" );
document.write( "medians intersect:\r\n" );
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document.write( "We set the two expressions for y equal:\r\n" );
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document.write( "Multiply through by 3\r\n" );
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document.write( "Substitute in\r\n" );
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document.write( "So the centroid (the point where the two (green)\r\n" );
document.write( "medians intersect is  \r\n" );
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document.write( "Edwin
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