Find the point on a directed line segment that partitions it in a given ratio (NC.M1.G-GPE.6).

What this topic is asking

NC.M1.G-GPE.6 asks you to find the point on a directed line segment between two given points that partitions it in a given ratio. "Directed" means the segment has a starting point and a direction (from $A$ to $B$ is different from $B$ to $A$), and the ratio tells you how the point splits the segment.

The ratio and the fraction

The ratio tells you the fraction of the way to travel.

The section method

Why the midpoint is the 1 to 1 case

The midpoint is just partitioning in ratio $1:1$, so the fraction is $\frac{1}{1+1} = \frac{1}{2}$. Using the section method with $\frac{1}{2}$ gives $\left(x_1 + \frac{1}{2}(x_2 - x_1),\ y_1 + \frac{1}{2}(y_2 - y_1)\right)$, which simplifies to the midpoint formula $\left(\frac{x_1 + x_2}{2},\ \frac{y_1 + y_2}{2}\right)$. So the midpoint is one special partition.

Partitioning a horizontal or vertical segment

When a segment is purely horizontal or vertical, one coordinate stays fixed and you only partition the other, which is a quick check that you understand the method. For a horizontal segment from $(2, 5)$ to $(14, 5)$ partitioned $1:2$ from the left end, the $y$-coordinate stays $5$ and you move $\frac{1}{3}$ of the horizontal change: $x = 2 + \frac{1}{3}(14 - 2) = 2 + 4 = 6$, so the point is $(6, 5)$. For a vertical segment, the $x$-coordinate stays fixed and you partition the $y$ values the same way. These cases confirm the section method without the distraction of both coordinates changing, and they show that the fraction $\frac{m}{m+n}$ applies independently to each coordinate.

How the NC Math 1 EOC examines this topic

• Gridded response. Find a coordinate of the partition point.
• Multiple choice. Choose the partition point, or identify the fraction for a given ratio.
• Calculator-active. Partition computations fit the calculator-active section.

This extends the midpoint idea and uses the same coordinate reasoning as coordinate proofs.

Why direction and order matter

The word "directed" is the trap and the point. Partitioning from $A$ to $B$ in ratio $1:3$ lands $\frac{1}{4}$ of the way from $A$, but partitioning from $B$ to $A$ in ratio $1:3$ lands $\frac{1}{4}$ of the way from $B$, a different point entirely. The ratio is always measured from the first named endpoint, so reading which point is the start is as important as the arithmetic. Thinking of it as "travel a fraction of the journey from the start" keeps this straight: you begin at $A$, head toward $B$, and stop after the fraction $\frac{m}{m+n}$ of the trip. This is also why the midpoint, exactly half the journey, is the cleanest special case.

Try this

Q1. Find the point $\frac{1}{2}$ of the way from $(0, 0)$ to $(8, 6)$. [1 point]

• Cue. Midpoint: $(4, 3)$.

Q2. Partition from $A(0, 0)$ to $B(12, 8)$ in ratio $3:1$ from $A$. [2 points]

• Cue. Fraction $\frac{3}{4}$: $x = \frac{3}{4}(12) = 9$, $y = \frac{3}{4}(8) = 6$, so $(9, 6)$.