Question

based on the diagram, can point d be the centroid of triangle acf? explain.
yes, point d is the point of intersection of segments drawn from all three vertices.
yes, de is three - quarters of the length of the full segment.
no, de should be longer than ad.
no, the ratio between ad and de is 3:1.

Explanation:

To determine if \( D \) is the centroid of \( \triangle ACF \), recall the centroid property: the centroid divides each median into a ratio of \( 2:1 \) (longer segment to shorter segment). For a median from a vertex to the midpoint of the opposite side, the distance from the vertex to centroid (\( AD \)) and centroid to midpoint (\( DE \)) should have a \( 2:1 \) ratio (or \( AD:DE = 2:1 \), meaning \( AD = 2 \times DE \)).

• Check \( AD = 12 \), \( DE = 4 \). The ratio \( AD:DE = 12:4 = 3:1 \), which does not match the centroid’s \( 2:1 \) ratio.
• Analyze other options:
• First option: Intersection of three segments is not enough (they must be medians, not just any segments).
• Second option: Centroid divides the median into \( 2:1 \), so \( DE \) should be \( \frac{1}{3} \) of the full segment (not \( \frac{3}{4} \)).
• Third option: \( DE \) is shorter than \( AD \), but the ratio (not just length) is incorrect.

Answer:

No, the ratio between AD and DE is 3:1.