Question

Triangle A B C with centroid G is shown on an x and y-axis. Lines are drawn from each point through the centroid to the opposite side to form line segments A F, B D, and C E. Point A is at (0, 0), point D is at (c, 0), point C is at (2 c, 0), point B is at (2 a, 2 b), point F is at (a + c, b),and point E is at (a, b).
The equation of line Line A Gis y = (StartFraction b Over a + c EndFraction)x. The midpoint of BC is (a + c, b). Does the midpoint of BC lie on Line A G? Why or why not?

no, because (StartFraction b Over a + c EndFraction)b does not equal a + c
no, because (StartFraction b Over a + c EndFraction)(a + c) does not equal b
yes, because (StartFraction b Over a + c EndFraction)b = a + c
yes, because (StartFraction b Over a + c EndFraction)(a + c) = b

Answer 1

Yes — because the midpoint of BC is (a + c, b). Substituting x = a + c into the line AG: y = (b/(a + c))(a + c) = b, so the point (a + c, b) lies on AG. Thus "yes, because (b/(a + c))(a + c) = b."