Finding trisection points on a line segment using coordinates

Key concept: This problem uses the midpoint idea to locate points that divide a segment into three equal parts.

Step by step

Core idea

A point of trisection divides a segment into three equal lengths. If you know the coordinates of the endpoints, you can find those points by using the same coordinate-averaging idea behind the midpoint formula, but applied to one-third and two-thirds of the way from one endpoint to the other.

For a segment from $A(x_1,y_1)$ to $B(x_2,y_2)$, the vector from $A$ to $B$ is $(x_2-x_1,\,y_2-y_1)$. One trisection point is found by moving one-third of that vector from $A$, and the other by moving two-thirds of that vector from $A$.

Step-by-step for the given points

Here, $A=(4,7)$ and $B=(16,-2)$.

First find the change in coordinates:

• $x$-change: $16-4=12$
• $y$-change: $-2-7=-9$

Now take one-third of each change:

• One-third of $12$ is $4$
• One-third of $-9$ is $-3$

So the first trisection point, $P$, is

$P=(4+4,\,7-3)=(8,4).$

For the second trisection point, $Q$, move two-thirds of the way from $A$ to $B$:

• Two-thirds of $12$ is $8$
• Two-thirds of $-9$ is $-6$

So

$Q=(4+8,\,7-6)=(12,1).$

Why this works

The midpoint formula gives the point halfway between two endpoints by averaging their coordinates. Trisection points work the same way, except the segment is divided into three equal parts instead of two. That means the first point is at $\tfrac13$ of the full displacement and the second point is at $\tfrac23$ of the full displacement.

This method is often faster than trying to use separate line equations, especially when the question gives direct coordinates.

Final answer

The trisection points are

$P=(8,4),\qquad Q=(12,1).$

Pitfall alert

A common mistake is to average the coordinates twice and think that gives the trisection points. The midpoint only gives the center of the segment, not points one-third and two-thirds along it. Another frequent error is to divide the endpoint coordinates themselves by 3 instead of dividing the displacement from one endpoint to the other. Always start from one endpoint, find the vector to the other endpoint, then scale that vector by $\tfrac13$ and $\tfrac23$.

Try different conditions

If the problem instead asked for the trisection points of the segment from $A=(-2,5)$ to $B=(10,11)$, the same method would apply. The displacement is $(12,6)$, so the points would be $(-2,5)+(4,2)=(2,7)$ and $(-2,5)+(8,4)=(6,9)$. Any trisection question with coordinates can be solved by the same one-third and two-thirds vector method.

Further reading

midpoint formula, segment trisection, coordinate geometry