Vectors
Hexagon centroid identity
MJMT_Full_Test_02
Grade 12
Question:
Let $ABCDEF$ be a hexagon with unequal side lengths. Let $G$ and $H$ be centroids of $\triangle ACE$ and $\triangle BDF$ respectively. If $\overrightarrow{AB}-\overrightarrow{BC}+\overrightarrow{CD}-\overrightarrow{DE}+\overrightarrow{EF}-\overrightarrow{FA}=\lambda\overrightarrow{GH}$, then $\lambda$ equals
Step-by-Step Solution
Key Concept: $G=(A+C+E)/3$, $H=(B+D+F)/3$. Compute $\overrightarrow{AB}-\overrightarrow{BC}+\cdots = 2(B+D+F)-2(A+C+E) = 6\overrightarrow{GH}$.
$\lambda=6$.
Correct Answer: 3