In a triangle ABC, medians AD and BE are perpendicular to each other, and have…

Concept: A triangle's centroid divides each median in a fixed 2:1 ratio (vertex-to-centroid : centroid-to-opposite-side's-midpoint), and it also splits the whole triangle into three smaller triangles of equal area — each exactly one-third of the total. When two medians are perpendicular, the two centroid-to-vertex segments become the legs of a right angle, which lets the whole area be built from that right triangle.

Application:

1. The centroid G divides each median in its fixed 2:1 ratio measured from the vertex, so AG = (2/3) × 12 = 8 cm (and GD = 4 cm); likewise BG = (2/3) × 9 = 6 cm (and GE = 3 cm).

2. Since AD is perpendicular to BE, the angle at G between GA and GB is 90°, so triangle AGB is right-angled at G with legs AG = 8 cm and GB = 6 cm.

3. Area(triangle AGB) = 1/2 × AG × GB = 1/2 × 8 × 6 = 24 sq cm.

4. The centroid always splits a triangle into three smaller triangles of equal area — triangle AGB, triangle BGC, and triangle CGA each equal one-third of the whole triangle. So Area(triangle ABC) = 3 × Area(triangle AGB) = 3 × 24 = 72 sq cm.

Cross-check:

Independently, for two perpendicular medians of lengths m1 and m2, the same centroid argument gives Area(triangle ABC) = (2/3) × m1 × m2 = (2/3) × 12 × 9 = 72 sq cm — matching the step-by-step result and confirming 72 sq cm as the area.