Question

Tracy found the length of GF. Her work is shown below.

Triangle A B C has centroid G. Lines are drawn from each point through the centroid to the midpoint of the opposite side to form line segments A E, B F, and C D. The length of line segment A G is 16, the length of line segment G E is 2 x, and the length of line segment B G is 8 x.

16 = 2(2x)
16 = 4x; x = 4
8(4) = 32
GF = 2(32)
GF = 64
What error did Tracy make?

She used AG = 2EG, but the correct statement is 2AG = GE.
She found that x = 4, but the correct value for x is 2.
She used GF = 2BG, but the correct statement is BG = 2GF.
She found that GF = 64, but the correct value of GF is 8.

Answer 1

Answer: She used GF = 2BG, but the correct statement is BG = 2GF.

Explanation: The centroid divides a median so the segment from the vertex to the centroid is twice the segment from the centroid to the midpoint. So BG = 2·GF, not GF = 2·BG. Tracy inverted that ratio (so she multiplied by 2 instead of dividing by 2).