Medians & Centroid
A median connects a vertex to the midpoint of the opposite side. The three medians always meet at one point — the centroid — the center of gravity.
Introduction
The centroid is the “balance point” of a triangle — if you cut the triangle out of cardboard, it would balance perfectly on a pin at the centroid. The centroid divides each median in a specific 2:1 ratio.
Past Knowledge
Midpoints (1.2). Midsegments (6.1.1). Centers (6.1.2–6.1.3).
Today's Goal
Locate the centroid and use the 2:1 ratio.
Future Success
Altitudes & orthocenter (6.1.5). Coordinate proofs. Physics balance.
Key Concepts
Centroid Theorem (2:1 Ratio)
The centroid divides each median into two segments in a 2:1 ratio. The longer segment is from the vertex to the centroid; the shorter is from the centroid to the midpoint.
Coordinate Centroid Formula
Given vertices , , :
Worked Examples
2:1 Ratio
Median has length 12. is the centroid. Find and .
,
Finding the Centroid
Find the centroid of with , , .
Working Backward from the Centroid
In , is the centroid on median . If , find and .
Centroid divides in 2:1 ratio. →
Common Pitfalls
Reversing the 2:1 Ratio
The longer part is from the vertex to the centroid (⅔ of the median). The shorter part is from centroid to midpoint (⅓). Don't flip them.
Median ≠ Altitude ≠ Perpendicular Bisector
A median goes to the midpoint of the opposite side. An altitude goes perpendicular to the opposite side. These are different unless the triangle is equilateral.
Real-Life Applications
Physics — Center of Mass
The centroid is the center of mass of a uniform triangular plate. This concept extends to engineering: compute the centroid to know where a crane should lift a triangular panel, or where a swimming pool cover will balance.
Urban Planning
When three communities form a triangle, the centroid represents the optimal location for a shared facility (park, water treatment plant) to minimize average distance to all three areas.
Practice Quiz
Loading...