Partitioning a Segment in a Given Ratio

From ratio to fraction

A ratio of $m:n$ cuts the segment into $m + n$ equal parts. Starting from $A$, the point $P$ sits at the end of the first $m$ parts, so $P$ is $\displaystyle \frac{m}{m+n}$ of the way from $A$ to $B$.

This is the step where most mistakes happen. A ratio of $2:3$ does not mean $\displaystyle \frac{2}{3}$ of the way — it means $\displaystyle \frac{2}{2+3} = \frac{2}{5}$ of the way. Always add both parts of the ratio to get the denominator. The diagram below cuts $\overline{AB}$ into $5$ equal parts, with $P$ landing after the first $2$.

The run-and-rise method

Call the fraction $\displaystyle k = \frac{m}{m+n}$. First find the run and the rise from $A(x_1, y_1)$ to $B(x_2, y_2)$: the run is $x_2 - x_1$ and the rise is $y_2 - y_1$. Then take $k$ of each and add to $A$.

As one formula: $P = \left(x_1 + k(x_2 - x_1),\; y_1 + k(y_2 - y_1)\right)$. You do not need to memorize it as a formula if you remember the idea — go $k$ of the way over, and $k$ of the way up, starting from $A$.

Direction matters

The ratio $AP:PB = 2:3$ and the ratio $BP:PA = 2:3$ give different points, because the counting starts from different ends. Always check which point the ratio starts from, and make that point your $(x_1, y_1)$.

A quick sanity check: if the fraction $k$ is less than $\displaystyle \frac{1}{2}$, the point should land closer to the starting point; if $k$ is more than $\displaystyle \frac{1}{2}$, closer to the far end. A ratio of $1:1$ gives $\displaystyle k = \frac{1}{2}$, which is exactly the midpoint.

Worked examples

Try one yourself

Common questions