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Partition of a directed line segment in a given ratio

20 practice questions 2 video lessons Theory + worked examples

Partitioning a Directed Segment

California Geometry • Standard G-GPE.6 • Coordinate Geometry

Partitioning a Directed Segment is a topic in Coordinate Geometry in the California Common Core State Standards. It is aligned to Standard G-GPE.6, which requires students to find the point on a directed line segment that partitions it in a given ratio.

Partitioning a directed segment in the ratio \(m:n\) places a point the fraction \(\dfrac{m}{m+n}\) of the way from one end to the other.

California Geometry › Coordinate Geometry › Partitioning a Directed Segment  —  Standard G-GPE.6

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Theory

A directed segment from \(A\) to \(B\) has a direction. The point \(P\) that partitions it in the ratio \(m:n\) (so \(AP:PB=m:n\)) lies the fraction \(k=\dfrac{m}{m+n}\) of the way from \(A\) to \(B\):

\[P=\big(x_1+k(x_2-x_1),\ y_1+k(y_2-y_1)\big).\]
Start at \(A\) and move a fraction of the way toward \(B\). The midpoint is the special case \(m:n=1:1\).
Partition in ratio 2:1 Point P divides segment AB so that AP to PB is two to one; P is two thirds of the way from A to B. A(1,1) B(7,4) P(5,3) 2 1
\(P\) divides \(AB\) in ratio \(2:1\), so \(P\) is \(\dfrac23\) of the way from \(A\).
Partition A to B in ratio m:n Partition A to B in ratio m:n Partition A to B in ratio m:n P = A + (m / (m+n)) · (B - A) x = x₁ + (m/(m+n))(x₂-x₁) y = y₁ + (m/(m+n))(y₂-y₁)
The partition formula.

The partition point:

\[k=\dfrac{m}{m+n},\qquad P=\big(x_1+k(x_2-x_1),\ y_1+k(y_2-y_1)\big)\]
the fraction is m over m plus n; add that fraction of the change to the starting coordinates
The ratio is directed: \(m\) is the part next to \(A\), so the fraction \(k\) is measured from \(A\).

How to partition a segment

  1. Identify the start \(A\), the end \(B\), and the ratio \(m:n\).
  2. Compute the fraction \(k=\dfrac{m}{m+n}\).
  3. Add \(k\) times the change to each starting coordinate.
  4. Check that \(P\) lies between \(A\) and \(B\).
Example 1 — Partition in ratio 2:1
Find the point that partitions the segment from \(A(1,1)\) to \(B(7,4)\) in the ratio \(2:1\) (from \(A\)).
Solution

The point is \(\dfrac{2}{2+1}=\dfrac23\) of the way from \(A\) to \(B\).

\(x\)\(=\)\(1+\dfrac23(7-1)=1+4=5\)
\(y\)\(=\)\(1+\dfrac23(4-1)=1+2=3\)
the point is five comma three
Example 2 — Ratio 1:3
Partition the segment from \((0,0)\) to \((8,12)\) in the ratio \(1:3\).
Solution

The fraction from the start is \(\dfrac{1}{1+3}=\dfrac14\).

\(x\)\(=\)\(0+\dfrac14(8-0)=2\)
\(y\)\(=\)\(0+\dfrac14(12-0)=3\)
the point is two comma three
Example 3 — Midpoint as a 1:1 partition
Show the ratio \(1:1\) gives the midpoint of \((2,4)\) and \((6,10)\).
Solution

The fraction is \(\dfrac{1}{1+1}=\dfrac12\).

\(x\)\(=\)\(2+\dfrac12(6-2)=4\)
\(y\)\(=\)\(4+\dfrac12(10-4)=7\)
the point is four comma seven, the midpoint
Example 4 — Ratio 3:2
Partition from \(A(-1,2)\) to \(B(9,7)\) in the ratio \(3:2\).
Solution

The fraction is \(\dfrac{3}{3+2}=\dfrac35\).

\(x\)\(=\)\(-1+\dfrac35(9-(-1))=-1+6=5\)
\(y\)\(=\)\(2+\dfrac35(7-2)=2+3=5\)
the point is five comma five

Common pitfalls

The fraction is \(\dfrac{m}{m+n}\), not \(\dfrac{m}{n}\).
Measure from the correct end — the ratio is directed from \(A\) to \(B\).
Apply \(k\) to the change \((x_2-x_1)\), then add the starting coordinate.

Frequently asked questions

How do you partition a segment in a given ratio?

Move the fraction \(k=\dfrac{m}{m+n}\) of the way from \(A\) to \(B\): \(P=(x_1+k(x_2-x_1),\,y_1+k(y_2-y_1))\).

What fraction of the way is a 2:1 partition?

\(\dfrac{2}{2+1}=\dfrac23\) of the way from the first point.

How is a midpoint related to partitioning?

The midpoint is the \(1:1\) partition, the fraction \(\dfrac12\).

Why is it called a directed segment?

Because the ratio is measured in a specific direction, from \(A\) toward \(B\); reversing the order changes the point.

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