JEE Advanced Maths: Properties of Triangles / Coordinate Geometry questions with solutions

Q1. In triangle ABC, the medians AD and BE intersect at right angles. If AD = 6 and BE = 9/2, find the area of triangle ABC.

1. 18
2. 27/2
3. 9
4. 36

Answer: 18

The centroid G divides AD and BE in ratio 2:1. So AG = (2/3)*6 = 4, GD = 2; BG = (2/3)*(9/2) = 3, GE = 3/2. Since AD perpendicular BE, the area of triangle ABG = (1/2)*AG*BG = (1/2)*4*3 = 6. A known result: when two medians are perpendicular, Area(ABC) = (2/3)*(product of the medians) = (2/3)*6*(9/2) = (2/3)*27 = 18. (Equivalently, Area(ABC) = 3 * Area(ABG) = 3*6 = 18.)