Partitioning a Segment
Find the point that divides a directed segment in a given ratio using the section formula. Apply to finding points that split a segment into equal thirds, or in a 2:3 ratio.
A lesson of Geometry, in Coordinate Geometry, about 10 minutes.
By the end of this lesson, you will be able to find the point that divides a directed line segment in a given ratio $m:n$, apply the section formula, and solve multi-step coordinate problems.
The Big Idea
Finding a point that divides a segment in a ratio is more precise than finding the midpoint (which is always 1:1). The directed segment concept — specifying direction from one endpoint to the other — determines a unique point for any ratio $m:n$.
The Partition Formula
To find point $P$ that divides $\overline{AB}$ from $A$ to $B$ in ratio $m:n$:
$$P = \left(x_A + \frac{m}{m+n}(x_B - x_A),; y_A + \frac{m}{m+n}(y_B - y_A)\right)$$
Interpretation: start at $A$, move $\dfrac{m}{m+n}$ of the way toward $B$.
Special case: $m:n = 1:1$ → midpoint formula.
Example 1 — Ratio 3:1
Find the point $P$ that divides $\overline{AB}$ in ratio $3:1$ from $A(1,2)$ to $B(9,6)$.
$$P_x = 1 + \frac{3}{4}(9-1) = 1 + 6 = 7$$
$$P_y = 2 + \frac{3}{4}(6-2) = 2 + 3 = 5$$
$P = (7, 5)$
Verify: $AP = \sqrt{36+9} = \sqrt{45} = 3\sqrt{5}$; $PB = \sqrt{4+1} = \sqrt{5}$. Ratio $= 3\sqrt{5} : \sqrt{5} = 3:1$ ✓
Example 2 — Ratio 2:3
Find the point $Q$ that divides $\overline{CD}$ from $C(-4,1)$ to $D(6,6)$ in ratio $2:3$.
$$Q_x = -4 + \frac{2}{5}(6-(-4)) = -4 + \frac{2}{5}(10) = -4 + 4 = 0$$
$$Q_y = 1 + \frac{2}{5}(6-1) = 1 + 2 = 3$$
$Q = (0, 3)$
Example 3 — Finding the Ratio
Point $M(2, 4)$ divides $\overline{AB}$ from $A(-2,0)$ to $B(6,8)$. Find the ratio $AM:MB$.
Using $x$-coordinates:
$$2 = -2 + \frac{m}{m+n}(6-(-2)) = -2 + \frac{8m}{m+n}$$
$$4 = \frac{8m}{m+n} \implies m+n = 2m \implies n = m$$
Ratio $m:n = 1:1$ — $M$ is the midpoint.
Check: Midpoint of $A(-2,0)$ and $B(6,8)$: $(2, 4)$ ✓
Example 4 — Trisecting a Segment
Trisect $\overline{PQ}$ where $P(0,0)$ and $Q(9,6)$. Find both trisection points.
First trisection point (ratio $1:2$ from $P$):
$$T_1 = \left(0 + \frac{1}{3}(9), 0 + \frac{1}{3}(6)\right) = (3, 2)$$
Second trisection point (ratio $2:1$ from $P$):
$$T_2 = \left(0 + \frac{2}{3}(9), 0 + \frac{2}{3}(6)\right) = (6, 4)$$
Example 5 — Real-World Application
A road runs straight from city $A(0,0)$ to city $B(20,15)$. A rest stop is located $\frac{3}{5}$ of the way from $A$ to $B$. Find its location.