📐 Concept: Medians and Centroid in a Triangle

In a triangle, the three medians (joining vertex to midpoint of opposite side) intersect at a point called the centroid, usually denoted as \( G \).

• Medians: \( AD, BE, CF \)
• Centroid \( G \) divides each median in the ratio \( 2:1 \)

That is,

\[ \frac{AG}{GD} = \frac{BG}{GE} = \frac{CG}{GF} = \frac{2}{1} \]

💡 Properties:

• The centroid always lies inside the triangle.
• It is the center of gravity for a uniform triangular plate.
• Each median divides the triangle into two triangles of equal area.

🧪 CAT-Style Question on Centroid

Q. \( \triangle ABC \) is a triangle and \( G \) is its centroid. If \( AG = BG \), what is \( \angle BGC \)?

📝 Solution:

We know from centroid property that \( AG:GD = 2:1 \), but here we are told \( AG = BG \).

This implies triangle symmetry such that:

- \( AG = BG \)
- \( D \) is the midpoint of \( BC \)
- \( AG \perp BC \)

\[ \Rightarrow \angle BGC = 90^\circ \]

✅ Final Answer: \( \boxed{90^\circ} \)