document.write( "Question 1185390: 3. The coordinates of vertex A of the triangle ABC are (7, -4). If the coordinates of the centroid of the triangle are (1, 2), find the coordinates of the midpoint of the side BC. \n" ); document.write( "

Algebra.Com's Answer #849629 by greenestamps(13200) $\"\"$ $\"About$

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\n" ); document.write( "Let D be the midpoint of side BC.

\n" ); document.write( "The three medians of any triangle meet at the centroid.

\n" ); document.write( "The point of intersection of the three medians divides each median into two parts whose lengths are in the ratio 2 to 1.

\n" ); document.write( "To state that differently, on each median, the distance from the vertex to the centroid is twice the distance from the centroid to the midpoint of the opposite side.

\n" ); document.write( "Using that, the problem is most easily solved informally, using logical reasoning and simple arithmetic.

\n" ); document.write( "Vertex A is at (7,-4); the centroid is at (1,2).

\n" ); document.write( "From the vertex to the centroid is a distance -6 in the x direction and +6 in the y direction, so from the centroid to midpoint D of side BC is half those distances: -3 in the x direction and +3 in the y direction.

\n" ); document.write( "-3 in the x direction and +3 in the y direction from the centroid (1,2) puts us at (-2,5).

\n" ); document.write( "ANSWER: (-2,5)

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