In triangle ABC, medians AD, BE, and CF intersect at centroid G. If area(ΔABC) = 36 cm^2, find area(ΔCGE).
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A12 sq cm
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B6 sq cm
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C9 sq cm
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D18 sq cm
Answer
Correct Answer: 6 sq cm
Explanation
Introduction / Context:Medians of a triangle intersect at the centroid, partitioning the triangle into six smaller triangles of equal area. Recognizing this partition lets us read off areas of sub-triangles like ΔCGE directly.
Given Data / Assumptions:
- AD, BE, CF are medians; E is midpoint of AC.
- G is the centroid (intersection of the three medians).
- Area(ΔABC) = 36 cm^2.
Concept / Approach:Centroid divides the triangle into six equal-area small triangles formed by the three medians. Thus each small triangle has area (total area)/6.
Step-by-Step Solution:
Each small sub-triangle area = 36 / 6 = 6 cm^2ΔCGE is one of these six equal small triangles.Therefore area(ΔCGE) = 6 cm^2Verification / Alternative check:Draw medians; around each vertex, two small triangles meet; all six are congruent in area by symmetry and midpoint properties.
Why Other Options Are Wrong:
- 9, 12, 18: These assume thirds or halves, not sixths.
Common Pitfalls:Forgetting that three medians create 6, not 4, parts; confusing centroid 2:1 segment property with area partition.
Final Answer:6 sq cm