Question

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

• A. 68
• B. 72
• C. 78
• D. 80

Answer 1

The Correct Option is B

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: \[ \text{Area}_{\triangle ABC} = 6 \times \text{Area}_{\triangle BGD} = 6 \times 12 = \boxed{72} \]