Overview

Medians intersect at the centroid, which splits each median in a $2:1$ ratio. This point is also the balance point of the triangle.

Key Ideas

• The centroid divides each median $2:1$ from the vertex.
• The three medians divide the triangle into 6 equal-area small triangles.
• Each median splits the triangle into two equal-area triangles.

Core Skills

Use the $2:1$ Ratio

If $G$ is the centroid and $M$ is the midpoint of a side, then $AG:GM=2:1$ on median $AM$.

Area Ratios

Medians create equal-area regions. Combine this with known areas to solve for unknowns quickly.

Midpoint Facts

By definition, each median hits the midpoint of a side, so segment halves are immediate and often appear in ratio problems.

Worked Example

In triangle $ABC$, the centroid $G$ lies on median $AM$. If $AG=10$, find $GM$.

The centroid ratio is $AG:GM = 2:1$, so $GM = 5$.

More Examples

Example 1: Full Median Length

If $AG=8$ on median $AM$, find $AM$.

$AG = \frac{2}{3}AM$, so $AM = 12$.

Example 2: Area Split

In triangle $ABC$, median $AM$ splits the triangle into two triangles of equal area. If $[ABC]=42$, find $[ABM]$.

$[ABM]=21$.

Example 3: Six Equal Areas

The medians divide a triangle into six equal areas. If one of the small triangles has area $5$, what is the area of the whole triangle?

$6\cdot 5 = 30$.

Strategy Checklist

• Mark midpoints first to locate medians.
• Use the $2:1$ ratio along medians to find segment lengths.
• Convert areas to ratios before using actual numbers.

Common Pitfalls

• Flipping the $2:1$ ratio.
• Assuming medians are perpendicular.
• Confusing medians with angle bisectors or altitudes.

Practice Problems