Centre of gravity of a triangle in engineering mechanics: The centroid is located at the intersection of which set of internal lines?

Introduction / Context:

The centroid (centre of gravity for a uniform lamina) of a triangle is a foundational result in statics and structural analysis. It is used to compute moments, locate resultant forces, and determine support reactions for triangular load distributions.

Given Data / Assumptions:

• Plane triangular lamina of uniform density.
• Standard Euclidean geometry applies.
• No skewed mass distribution or thickness variation.

Concept / Approach:

A triangle has several notable concurrent line sets: medians, angle bisectors, perpendicular bisectors, and altitudes. The point where medians meet is called the centroid (also denoted G). Each median connects a vertex to the midpoint of the opposite side. The centroid divides each median in a 2:1 ratio measured from the vertex.

Step-by-Step Solution:

Verification / Alternative check:

Coordinate geometry check: For vertices (x1,y1), (x2,y2), (x3,y3), the centroid is at ((x1 + x2 + x3)/3, (y1 + y2 + y3)/3), confirming the intersection of medians interpretation.

Why Other Options Are Wrong:

• Perpendicular bisectors meet at the circumcenter, not the centroid.
• Angle bisectors meet at the incenter (centre of inscribed circle).
• Altitudes meet at the orthocenter.
• “None of these” is incorrect because medians are the right answer.

Common Pitfalls:

• Confusing different triangle centers: centroid, incenter, circumcenter, and orthocenter.

Final Answer:

medians of the triangle