Partitioning a Segment in a Given Ratio

Quick answer

To find the point that splits a segment from A to B in the ratio m:n, turn the ratio into a fraction of the whole: m out of m + n parts. Then go that fraction of the way from A, which means multiplying the step from A to B by the fraction and adding it to A. It works because the run and rise shrink together, the same way a dilation from A would shrink them.

What you'll learn

Convert a partition ratio into a fraction of the segment
Find the point that partitions a directed segment in a given ratio
Explain why the method works using similar triangles

A point part of the way along

Find the point one third of the way from $A(1, 2)$ to $B(10, 8)$ .

The step from $A$ to $B$ is $9$ right and $6$ up. A third of that step is $3$ right and $2$ up. Take that much of the step, starting from $A$ :

P = (1 + 3,\; 2 + 2) = (4, 4)

The point one third of the way from A to B

$P$ splits the segment into $AP$ and $PB$ . $AP$ is one part and $PB$ is two, so $P$ partitions $AB$ in the ratio $1:2$ .

From a ratio to a fraction

A ratio compares the two pieces with each other. The method needs the fraction of the whole segment that lies before the point.

A ratio $m:n$ splits the segment into $m + n$ equal parts, and the point has $m$ of them behind it:

t = \frac{m}{m + n}

Ratio from $A$	Parts	Fraction $t$
$1:1$	$2$	$\tfrac{1}{2}$ , the midpoint
$1:2$	$3$	$\tfrac{1}{3}$
$2:3$	$5$	$\tfrac{2}{5}$
$3:1$	$4$	$\tfrac{3}{4}$

Then go that fraction of the step:

P = A + t(B - A) \qquad\text{so}\qquad x = x_1 + t(x_2 - x_1), \quad y = y_1 + t(y_2 - y_1)

Why scaling the step works

Look at the dashed triangles in the figure. The large one has legs $9$ and $6$ under $AB$ . The small one has legs $3$ and $2$ under $AP$ .

Both triangles have a right angle, and they share the angle at $A$ , so they are similar. Similar triangles scale every side by the same factor. Taking a third of the horizontal leg therefore forces a third of the vertical leg, and $P$ lands exactly on the segment, a third of its length from $A$ .

This is a dilation in disguise. $P$ is the image of $B$ under a dilation from center $A$ with scale factor $t$ : the step from $A$ to $B$ multiplied by $t$ and added back to $A$ . A dilation multiplies every length from its center by $t$ , so $AP$ is exactly $t$ times $AB$ .

The distances confirm it:

AP = \sqrt{3^2 + 2^2} = \sqrt{13} \qquad PB = \sqrt{6^2 + 4^2} = 2\sqrt{13}

$AP : PB = 1 : 2$ , as required.

Direction matters

A directed segment has a start. The ratio $1:2$ from $A$ to $B$ puts the point a third of the way from $A$ . The same ratio from $B$ to $A$ puts it a third of the way from $B$ , which is two thirds of the way from $A$ .

Segment	Ratio	Point
$A(1, 2)$ to $B(10, 8)$	$1:2$	$(4, 4)$
$A(1, 2)$ to $B(10, 8)$	$2:1$	$(7, 6)$

Always start the step at the first-named point.

Worked examples

Common mistakes

Practice problems

Find the point $\tfrac{1}{4}$ of the way from $(0, 0)$ to $(8, 12)$ .

Answer

$(2, 3)$

Full solution

A quarter of the step $(8, 12)$ is $(2, 3)$ , added to $(0, 0)$ .
Partition $A(0, 0)$ to $B(8, 4)$ in the ratio $3:1$ .

Answer

$(6, 3)$

Full solution

$t = \tfrac{3}{4}$ , and three quarters of the step $(8, 4)$ is $(6, 3)$ .
Partition the segment from $B(8, 4)$ to $A(0, 0)$ in the ratio $3:1$ .

Answer

$(2, 1)$

Full solution

Start at $B$ . The step to $A$ is $8$ left and $4$ down, and three quarters of it is $6$ left and $3$ down.

$(8 - 6,\; 4 - 3) = (2, 1)$ . Reversing the direction moved the point from $(6, 3)$ to $(2, 1)$ .
Find the midpoint of $(3, -5)$ and $(9, 1)$ .

Answer

$(6, -2)$

Full solution

Average the coordinates: $\tfrac{3 + 9}{2} = 6$ and $\tfrac{-5 + 1}{2} = -2$ .
Partition $A(2, 1)$ to $B(14, 9)$ in the ratio $1:3$ .

Answer

$(5, 3)$

Full solution

$t = \tfrac{1}{4}$ . The step is $(12, 8)$ , and a quarter of it is $(3, 2)$ .
Partition $A(-1, 4)$ to $B(9, -6)$ in the ratio $3:2$ .

Answer

$(5, -2)$

Full solution

$t = \tfrac{3}{5}$ . The step is $10$ right and $10$ down; three fifths is $6$ right and $6$ down.

$(-1 + 6,\; 4 - 6) = (5, -2)$ .
Find the point $\tfrac{3}{4}$ of the way from $-8$ to $4$ on a number line.

Answer

$1$

Full solution

The step is $12$ , and three quarters of it is $9$ . $-8 + 9 = 1$ .
Explain why the ratio $1:2$ puts the point one third of the way along.

Answer

The segment has $1 + 2 = 3$ equal parts, and the point has one of them before it.

Full solution

A ratio $1:2$ says the first piece is one part and the second is two parts of the same size. Together that is three parts, so each part is a third of the segment.

The point sits after the first part, one third of the way from the start.
Check with distances that $P(4, 4)$ partitions $A(1, 2)$ to $B(10, 8)$ in the ratio $1:2$ .

Answer

$AP = \sqrt{13}$ and $PB = 2\sqrt{13}$ , a ratio of $1:2$ .

Full solution

$AP = \sqrt{3^2 + 2^2} = \sqrt{13}$ and $PB = \sqrt{6^2 + 4^2} = \sqrt{52} = 2\sqrt{13}$ .

$\tfrac{AP}{PB} = \tfrac{1}{2}$ ✓
Asked to partition $A(1, 2)$ to $B(10, 8)$ in the ratio $1:2$ , Rosa uses $t = \tfrac{1}{2}$ and gets $(5.5, 5)$ . Find her error.

Hint

How many parts does the ratio $1:2$ describe?

Answer

She read the ratio as a fraction. The fraction is $\tfrac{1}{3}$ , and the point is $(4, 4)$ .

Full solution

$t = \tfrac{1}{2}$ gives the midpoint, which partitions the segment $1:1$ , not $1:2$ .

The ratio $1:2$ has three parts in all, so $t = \tfrac{1}{3}$ . A third of the step $(9, 6)$ is $(3, 2)$ , giving $(4, 4)$ .

Her point splits the segment into two equal halves, and the question asked for pieces in the proportion one to two.

Frequently asked questions

What does it mean to partition a segment in the ratio 1:2?

To find the point that splits it into two parts whose lengths compare as 1 to 2. That point is one third of the way from the start.

How do I turn a ratio into a fraction?

Add the parts. A ratio of m:n has m + n parts in all, and the point has m of them behind it, so it sits m/(m + n) of the way along.

What is a directed segment?

A segment with a start and an end. From A to B is not the same as from B to A, so the ratio 1:2 lands in a different place depending on the direction.

What is the formula?

P = A + t(B − A), where t = m/(m + n). In coordinates, x = x₁ + t(x₂ − x₁) and y = y₁ + t(y₂ − y₁).

How is the midpoint related?

The midpoint partitions a segment in the ratio 1:1, so t = 1/2, which averages the coordinates.

What to learn next

The Parabola from Its Focus and DirectrixWhy a parabola is every point equally far from a point and a line, how to derive its equation from that definition with the distance formula, and what the focus does.

Standards alignment

This lesson covers the following Common Core State Standards for Mathematics.

CCSS.MATH.CONTENT.HSG.GPE.B.6Expressing Geometric Properties with EquationsFind the point on a directed line segment between two given points that partitions the segment in a given ratio.

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