In triangle ABC, AD, BE are medians and G is the centroid. ∠AGE = 30°, AD =12, BE =18. Find the area of the triangle.
Explanation:
G is the centroid of ∆ABC and centroid divides the median in the ratio of 2 : 1.
∴ G divided AD in the ratio of 2 : 1 ⇒ AG = 2/3 × 12 = 8.
∴ G divided BE in the ratio of 2 : 1 ⇒ GE = 1/3 × 18 = 6.
⇒ Area of ∆ AGE = ½ × AG × GE × Sin30° = ½ × 8 × 6 × ½ = 12.
We also know that centroid divides the triangle in 6 smaller triangles of equal areas.
∴ Area(∆ABC) = 6 × Area(∆AGE) = 6 × 12 = 72.
Hence, option (b).
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