Question

Centroid of triangle lies at:

• A. Midpoint
• B. Intersection of medians
• C. Circumcenter
• D. Vertex

Hint:

The centroid is the point where a triangular cardboard sheet can be perfectly balanced on the tip of a needle.
It always lies inside the triangle.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks for the geometric definition of the centroid of a triangle.

Step 2: Key Formula or Approach:
In geometry and mechanics, the centroid of a shape represents its geometric center. For a thin plate of uniform density, the centroid coincides with its center of gravity.

Step 3: Detailed Explanation:

• A median of a triangle is defined as a straight line segment drawn from any vertex to the midpoint of the opposite side.

• Every triangle has exactly three medians.

• According to geometric principles, all three medians of a triangle always intersect at a single, common point.

• This point of concurrency is called the centroid of the triangle.

• The centroid divides each median in a $2:1$ ratio, with the larger part being closer to the vertex.

• Mathematically, if the vertices of a triangle are $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$, the coordinates of the centroid are:
\[ \bar{x} = \frac{x_1 + x_2 + x_3}{3}, \quad \bar{y} = \frac{y_1 + y_2 + y_3}{3} \]

Step 4: Final Answer:
The centroid of a triangle lies at the intersection of its medians.