JEE Advanced Maths: Triangles / Coordinate Geometry questions with solutions

Q1. In a right triangle ABC, right-angled at B, it can be shown that the distance between the circumcentre and the centroid satisfies GD = (1/3)(BD) = (1/3)(AC/2), where D is the midpoint of the hypotenuse AC. If AC = 10, find GD.

1. 1
2. 5/3
3. 3
4. 10/3

Answer: 5/3

In a right triangle, the circumcentre is the midpoint of the hypotenuse, so BD = AC/2 = 5 (the median to the hypotenuse equals half the hypotenuse). Using the given relation GD = (1/3)(AC/2) = (1/3)(5) = 5/3.

Q2. In triangle ABC with centroid G, the median segment AG equals the side BC. Find the measure of angle BGC (in degrees).

1. 90 deg
2. 60 deg
3. 120 deg
4. 150 deg

Answer: 90 deg

Let M be the midpoint of BC. The condition AG = BC together with the centroid ratios forces GM (the median from G in triangle BGC) to equal half of BC, which is the well-known condition for a right angle at G.