Medians & the Centroid

Q: How do I remember which piece is two-thirds?

The vertex piece is the long one: $\frac{2}{3}$ of the median from the vertex to the centroid, $\frac{1}{3}$ from the centroid to the midpoint. A quick check: the two pieces must add up to the whole median.

A median of a triangle is a segment that connects a vertex to the midpoint of the opposite side. Every triangle has three medians, and they always meet at a single point called the centroid. The centroid is the triangle's balance point — cut the triangle out of cardboard and it balances perfectly on a pin placed at the centroid.

The fact you will use over and over is the ratio. The centroid divides each median into two pieces, and it never cuts the median in half: the piece from the vertex to the centroid is twice as long as the piece from the centroid to the midpoint. Said another way, the vertex piece is $\displaystyle \frac{2}{3}$ of the whole median and the midpoint piece is $\displaystyle \frac{1}{3}$ .

The 2:1 ratio

Suppose $\overline{AM}$ is a median of $\triangle ABC$ and $P$ is the centroid. Three equivalent statements describe where $P$ sits: $\displaystyle AP = \frac{2}{3} \cdot AM$ , $\displaystyle PM = \frac{1}{3} \cdot AM$ , and $AP = 2 \cdot PM$ . Pick whichever one connects the number you know to the number you want.

Keep the pieces straight by remembering which one is longer: the piece touching the vertex is the long piece. If you ever compute a vertex piece that is shorter than the midpoint piece, you have the ratio upside down.

Given the whole median, multiply by $\displaystyle \frac{2}{3}$ or $\displaystyle \frac{1}{3}$ . Given one piece, use $AP = 2 \cdot PM$ to find the other, then add the pieces if you need the whole median.

The centroid on the coordinate plane

When a triangle's vertices are given as coordinates, the centroid is just the average of the three vertices: average the three $x$ -coordinates to get the centroid's $x$ , and average the three $y$ -coordinates to get its $y$ . No medians need to be drawn at all.

For the triangle below with vertices at $(-3, -2)$ , $(1, 4)$ , and $(5, 1)$ , the centroid is at $\left(\dfrac{-3 + 1 + 5}{3}, \dfrac{-2 + 4 + 1}{3}\right) = (1, 1)$ .

Median, or something else?

A median is defined by where it starts and where it ends: a vertex on one end, the midpoint of the opposite side on the other. It does not have to be perpendicular to anything. A segment from a vertex that meets the opposite side at a right angle is an altitude; a line that cuts a side at its midpoint at a right angle is a perpendicular bisector. On a test, check the tick marks — midpoint tick marks with no right-angle mark means median.

Worked examples

Example 1: from the median to the vertex piece

In $\triangle ABC$ , $\overline{AM}$ is a median and $P$ is the centroid. If $AM = 21$ , find $AP$ .

The vertex piece is two-thirds of the median

AP = \dfrac{2}{3} \cdot AM

Substitute

AP = \dfrac{2}{3}(21)

Simplify

AP = 14

Answer: $AP = 14$

Example 2: from one piece to the other

$P$ is the centroid of $\triangle DEF$ on median $\overline{DM}$ , and $PM = 6$ . Find $DP$ and the length of the whole median.

The vertex piece is twice the midpoint piece

DP = 2 \cdot PM

Substitute

DP = 2(6) = 12

Add the pieces for the whole median

DM = 12 + 6 = 18

Answer: $DP = 12$ and $DM = 18$

Example 3: centroid from coordinates

A triangle has vertices $(-3, -2)$ , $(1, 4)$ , and $(5, 1)$ . Find the centroid.

Average the

x

-coordinates

x = \dfrac{-3 + 1 + 5}{3} = 1

Average the

y

-coordinates

y = \dfrac{-2 + 4 + 1}{3} = 1

Write the centroid as a point

(1, 1)

Answer: $(1, 1)$

Try one yourself

Common questions

Does the centroid cut each median in half?

No — that is the classic trap. The centroid splits every median in a $2:1$ ratio. The piece from the vertex to the centroid is the long piece, twice the length of the piece from the centroid to the midpoint.

How do I remember which piece is two-thirds?

The vertex piece is the long one: $\displaystyle \frac{2}{3}$ of the median from the vertex to the centroid, $\displaystyle \frac{1}{3}$ from the centroid to the midpoint. A quick check: the two pieces must add up to the whole median.

Why is the centroid called the balance point?

It is the center of mass of the triangle. A flat triangle of uniform material balances on a support placed exactly at the centroid, which is why questions about balancing a tabletop or a mobile are asking for the centroid.

Is a median the same as a perpendicular bisector?

No. Both pass through a side's midpoint, but a median starts at the opposite vertex and usually is not perpendicular to the side, while a perpendicular bisector must form a right angle with the side and usually misses the opposite vertex.

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