In triangle ABC, medians AD, BE and CF intersect at centroid G. If the area of triangle ABC is 156 sq cm, what is the area (in sq cm) of the small triangle FGE formed by two medians and the centroid?

Introduction / Context:
This question checks understanding of medians and the centroid of a triangle, and how they divide the area of the triangle into smaller equal parts. Instead of using coordinates or long algebra, the key idea is to use well known area ratios created by medians and the centroid. This is a standard geometry concept that appears frequently in aptitude and school level competitive exams.

Given Data / Assumptions:

• In triangle ABC, AD, BE and CF are medians.
• All three medians intersect at a common point G, the centroid.
• The area of triangle ABC is 156 sq cm.
• We need the area of triangle FGE, formed by centroid G and two midpoints E and F on the sides.
• Triangle is assumed to be non degenerate and medians are drawn correctly.

Concept / Approach:
Each median of a triangle divides the triangle into two smaller triangles of equal area. When all three medians are drawn, they intersect at the centroid and divide the triangle into six smaller triangles of equal area. Points E and F are midpoints of sides, so triangle FGE is one of these six equal area regions. Therefore, its area is one sixth of the area of triangle ABC. Once this ratio is known, the calculation is very short.

Step-by-Step Solution:
Step 1: Recall that a median divides a triangle into two equal area parts. Step 2: When all three medians are drawn, they intersect at the centroid G, dividing the triangle into 6 small triangles of equal area. Step 3: Triangle FGE is one of these six small triangles inside triangle ABC. Step 4: Therefore, area of triangle FGE = (1 / 6) * area of triangle ABC. Step 5: Compute (1 / 6) * 156 = 156 / 6 = 26, then note that this is the area of each small triangle formed by medians and sides. Step 6: Observe carefully: triangle FGE is formed by joining midpoints and centroid, and it actually occupies half of one of those small regions, giving area = 26 / 2 = 13 sq cm.

Verification / Alternative check:
A quick coordinate geometry check can be done by placing triangle ABC with convenient coordinates and computing areas using determinants. Using such a model, one finds that triangle FGE indeed has area equal to one twelfth of the total area of triangle ABC. Since the given total area is 156 sq cm, division by 12 again confirms 13 sq cm for triangle FGE. So the answer is consistent across both area ratio reasoning and coordinate methods.

Why Other Options Are Wrong:
Option 26 corresponds to one sixth of the total area and ignores that triangle FGE is smaller than each median created triangle. Options 39 and 52 are larger fractions of the whole triangle and do not match the internal division created by medians. Option 78 would be half of the whole triangle and is clearly impossible for such a small internal region. Only 13 sq cm correctly reflects the precise fraction formed by F, G and E.

Common Pitfalls:
Learners often remember that medians create six equal regions but misidentify which region corresponds to the requested triangle. Another common error is to take one sixth of the area and stop, without checking the location and boundaries of FGE. Drawing a clear diagram and marking all six small triangles helps avoid confusion. Always relate the requested region back to these standard subdivisions before final calculation.

Final Answer:
The area of triangle FGE is 13 sq cm.