In a triangle ABC, medians AD and BE are perpendicular to each other, and have lengths 12 cm and 9 cm, respectively. Then, the area of triangle ABC, in sq cm, is
- 68
- 72
78
- 80
The Correct Option is B
Solution and Explanation
From the figure 
Let us consider a triangle ABC with medians AD, BE, and CF.
The medians of a triangle intersect at a point called the centroid, denoted by G.
Key Property: The centroid divides each median in the ratio 2:1, with the longer segment being from vertex to centroid.
Given:
- Length of median \( AD = 12 \)
- Length of median \( BE = 9 \)
Centroid Division:
The centroid divides median \( AD \) in the ratio \( 2:1 \):
\[ GD = \frac{1}{3} \times AD = \frac{1}{3} \times 12 = 4 \]
Similarly, centroid divides median \( BE \) in the ratio \( 2:1 \):
\[ GB = \frac{2}{3} \times BE = \frac{2}{3} \times 9 = 6 \]
Area of Triangle BGD:
\[ \text{Area}_{\triangle BGD} = \frac{1}{2} \times GB \times GD = \frac{1}{2} \times 6 \times 4 = 12 \]
Total Area of Triangle ABC:
It is a well-known geometric result that all three medians divide triangle \( ABC \) into six smaller triangles of equal area.
Hence, the total area is: \[ \text{Area}_{\triangle ABC} = 6 \times \text{Area}_{\triangle BGD} = 6 \times 12 = \boxed{72} \]
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