Partitioning a Line Segment: Ratio Formula

Partitioning a line segment means finding the point that splits a segment into two parts with a given ratio. To split $\overline{AB}$ in the ratio $a:b$ from $A$ , find the fraction $k = \frac{a}{a + b}$ . Then move that fraction of the way from $A$ to $B$ .

Before this lesson: You should know the Midpoint Formula, Ratios and Slope.

What Does It Mean to Partition a Segment?

A midpoint splits a segment into two equal parts. That is the ratio $1:1$ . But you can split a segment in other ratios too.

Say point $P$ splits $\overline{AB}$ in the ratio $1:3$ . Then $AP$ is 1 part and $PB$ is 3 parts. The whole segment is $1 + 3 = 4$ parts. So $P$ is $\frac{1}{4}$ of the way from $A$ to $B$ .

A segment with a starting point and an ending point is called a directed segment. The order matters. The ratio $1:3$ from $A$ lands near $A$ . From $B$ , it lands near $B$ .

A bar split into 5 equal parts. The first 2 parts are shaded blue and labeled 2 parts. The last 3 parts are shaded orange and labeled 3 parts. Point P is marked where the colors meet. Below: ratio 2:3 means 5 parts in all, so P is 2/5 of the way from A to B. — The ratio 2:3 cuts the segment into 2 + 3 = 5 equal parts. P is 2 of those parts from A, which is 2/5 of the way.

Partitioning a Line Segment Formula

To split the segment from $A(x_1, y_1)$ to $B(x_2, y_2)$ in the ratio $a:b$ , first find $k$ .

k = \frac{a}{a + b}

Then move $k$ of the run and $k$ of the rise from $A$ .

P = \big(x_1 + k(x_2 - x_1),\ y_1 + k(y_2 - y_1)\big)

The run is $x_2 - x_1$ and the rise is $y_2 - y_1$ , the same parts you use for slope.

Steps to partition a segment

Find the total number of parts: $a + b$ .
Find the fraction $k = \frac{a}{a + b}$ .
Find the run, $x_2 - x_1$ , and the rise, $y_2 - y_1$ .
Multiply the run and the rise by $k$ .
Add the results to the coordinates of $A$ .
Check: $P$ should sit on the segment, closer to $A$ when $a$ is less than $b$ .

Worked Examples

Example 1: Ratio 2:3

Find the point $P$ that partitions $\overline{AB}$ in the ratio $2:3$ , with $A(0, 0)$ and $B(10, 5)$ .

Step 1: Find the total number of parts.

2 + 3 = 5

Step 2: Find the fraction.

k = \frac{2}{5}

Step 3: Find the run.

10 - 0 = 10

Step 4: Find the rise.

5 - 0 = 5

Step 5: Multiply the run by $k$ (see multiplying fractions).

\frac{2}{5} \cdot 10 = 4

Step 6: Multiply the rise by $k$ .

\frac{2}{5} \cdot 5 = 2

Step 7: Add to the coordinates of $A$ .

P = (0 + 4,\ 0 + 2) = (4, 2)

So $P = (4, 2)$ .

Example 2: Ratio 1:3

Find the point $P$ that partitions $\overline{AB}$ in the ratio $1:3$ , with $A(1, 2)$ and $B(9, 14)$ .

Step 1: Find the total number of parts.

1 + 3 = 4

Step 2: Find the fraction.

k = \frac{1}{4}

Step 3: Find the run.

9 - 1 = 8

Step 4: Find the rise.

14 - 2 = 12

Step 5: Multiply the run by $k$ .

\frac{1}{4} \cdot 8 = 2

Step 6: Multiply the rise by $k$ .

\frac{1}{4} \cdot 12 = 3

Step 7: Add to the coordinates of $A$ .

P = (1 + 2,\ 2 + 3) = (3, 5)

So $P = (3, 5)$ . The graph below shows $P$ one quarter of the way from $A$ .

A coordinate grid from 0 to 10 across and 0 to 15 up. Segment AB runs from A(1, 2) to B(9, 14). Tick marks split it into 4 equal parts. Point P(3, 5) is the first tick mark from A, marked with an orange dot. The part from A to P is labeled 1 part, and the part from P to B is labeled 3 parts. — The ratio 1:3 splits AB into 4 equal parts. P(3, 5) is 1 part from A and 3 parts from B.

Example 3: Negative coordinates

Find the point $P$ that partitions $\overline{AB}$ in the ratio $2:1$ , with $A(-4, 6)$ and $B(5, -3)$ .

Step 1: Find the total number of parts.

2 + 1 = 3

Step 2: Find the fraction.

k = \frac{2}{3}

Step 3: Find the run. Subtracting a negative means adding (see adding and subtracting integers).

5 - (-4) = 9

Step 4: Find the rise.

-3 - 6 = -9

Step 5: Multiply the run by $k$ .

\frac{2}{3} \cdot 9 = 6

Step 6: Multiply the rise by $k$ .

\frac{2}{3} \cdot (-9) = -6

Step 7: Add to the coordinates of $A$ .

P = (-4 + 6,\ 6 + (-6)) = (2, 0)

So $P = (2, 0)$ . It is closer to $B$ , because the first part is the bigger one.

Example 4: A fraction answer

Find the point $P$ that partitions $\overline{AB}$ in the ratio $3:2$ , with $A(2, -1)$ and $B(7, 3)$ .

Step 1: Find the total number of parts.

3 + 2 = 5

Step 2: Find the fraction.

k = \frac{3}{5}

Step 3: Find the run.

7 - 2 = 5

Step 4: Find the rise.

3 - (-1) = 4

Step 5: Multiply the run by $k$ .

\frac{3}{5} \cdot 5 = 3

Step 6: Multiply the rise by $k$ .

\frac{3}{5} \cdot 4 = \frac{12}{5}

Step 7: Add the x-values.

2 + 3 = 5

Step 8: Add the y-values. Write $-1$ as $-\frac{5}{5}$ (see adding and subtracting fractions).

-\frac{5}{5} + \frac{12}{5} = \frac{7}{5}

Step 9: Write the point.

P = \left(5, \frac{7}{5}\right)

So $P = \left(5, \frac{7}{5}\right)$ , or $(5, 1.4)$ .

Example 5: Word problem

A hiking trail runs straight from a parking lot at $(-2, 1)$ to a lookout at $(10, 9)$ on a map. Each unit is 1 kilometer. A rest stop sits $\frac{3}{4}$ of the way from the parking lot to the lookout. Where is it?

Step 1: The fraction is given, so $k = \frac{3}{4}$ . (This is the ratio $3:1$ .)

Step 2: Find the run.

10 - (-2) = 12

Step 3: Find the rise.

9 - 1 = 8

Step 4: Multiply the run by $k$ .

\frac{3}{4} \cdot 12 = 9

Step 5: Multiply the rise by $k$ .

\frac{3}{4} \cdot 8 = 6

Step 6: Add to the parking lot’s coordinates.

P = (-2 + 9,\ 1 + 6) = (7, 7)

The rest stop is at $(7, 7)$ on the map.

Common Mistakes

Mistake	Example of the error	Fix
Using $\frac{a}{b}$ as the fraction	Ratio $2:3$ gives $k = \frac{2}{3}$	Divide by the total parts: $k = \frac{2}{2 + 3} = \frac{2}{5}$ .
Starting from the wrong end	Moving $\frac{1}{4}$ of the way from $B$ when the ratio is from $A$	Start at the first point named. “From $A$ to $B$ ” means start at $A$ .
Forgetting to add $A$ ‘s coordinates	Answering $(2, 3)$ in Example 2	That is only the move. Add it to $A(1, 2)$ to get $(3, 5)$ .
Multiplying the coordinates instead of the run and rise	Using $\frac{1}{4} \cdot 9$ for $B(9, 14)$	Multiply the run, $9 - 1 = 8$ , not the coordinate 9.
Sign errors in the rise	Writing $-3 - 6 = 3$	$-3 - 6 = -9$ . A negative rise means $P$ is lower than $A$ .

Partitioning on a Number Line and the Midpoint

On a number line, use the same idea with one coordinate. To split the segment from $-3$ to $12$ in the ratio $1:2$ :

k = \frac{1}{1 + 2} = \frac{1}{3}

12 - (-3) = 15

\frac{1}{3} \cdot 15 = 5

-3 + 5 = 2

The point is at $2$ . It is 5 units from $-3$ and 10 units from $12$ , and $5:10 = 1:2$ .

The ratio $1:1$ gives $k = \frac{1}{2}$ , which is the midpoint. So the midpoint formula is a special case of partitioning. You can also check a partition point with the distance formula: the two parts should have lengths in the ratio $a:b$ .

Practice Problems

Partition $\overline{AB}$ in the ratio $1:2$ with $A(0, 0)$ and $B(6, 9)$ .
Partition $\overline{AB}$ in the ratio $2:3$ with $A(2, 1)$ and $B(12, 6)$ .
Partition $\overline{AB}$ in the ratio $3:1$ with $A(-5, -2)$ and $B(3, 10)$ .
Partition $\overline{AB}$ in the ratio $1:2$ with $A(4, 7)$ and $B(-2, -2)$ .
On a number line, find the point that splits the segment from $-8$ to $16$ in the ratio $3:5$ .
Partition $\overline{AB}$ in the ratio $1:3$ with $A(1, -3)$ and $B(9, 1)$ .
Partition $\overline{AB}$ in the ratio $2:3$ with $A(-1, 2)$ and $B(4, 4)$ .
Partition $\overline{AB}$ in the ratio $1:1$ with $A(-6, 4)$ and $B(2, -8)$ . What is another name for this point?
Point $P(2, 1)$ partitions $\overline{AB}$ in the ratio $1:3$ . If $A$ is $(0, 0)$ , find $B$ .
A straight road runs from town $A(-3, -4)$ to town $B(9, 12)$ on a map. A gas station is $\frac{1}{4}$ of the way from $A$ to $B$ . Where is it?

Show answers

Total parts: $1 + 2 = 3$ . Fraction: $k = \frac{1}{3}$ . Run: $6 - 0 = 6$ . Rise: $9 - 0 = 9$ . Multiply the run: $\frac{1}{3} \cdot 6 = 2$ . Multiply the rise: $\frac{1}{3} \cdot 9 = 3$ . Add to A: $(0 + 2,\ 0 + 3) = (2, 3)$ . So $P = (2, 3)$ .
Total parts: $2 + 3 = 5$ . Fraction: $k = \frac{2}{5}$ . Run: $12 - 2 = 10$ . Rise: $6 - 1 = 5$ . Multiply the run: $\frac{2}{5} \cdot 10 = 4$ . Multiply the rise: $\frac{2}{5} \cdot 5 = 2$ . Add to A: $(2 + 4,\ 1 + 2) = (6, 3)$ . So $P = (6, 3)$ .
Total parts: $3 + 1 = 4$ . Fraction: $k = \frac{3}{4}$ . Run: $3 - (-5) = 8$ . Rise: $10 - (-2) = 12$ . Multiply the run: $\frac{3}{4} \cdot 8 = 6$ . Multiply the rise: $\frac{3}{4} \cdot 12 = 9$ . Add to A: $(-5 + 6,\ -2 + 9) = (1, 7)$ . So $P = (1, 7)$ .
Total parts: $1 + 2 = 3$ . Fraction: $k = \frac{1}{3}$ . Run: $-2 - 4 = -6$ . Rise: $-2 - 7 = -9$ . Multiply the run: $\frac{1}{3} \cdot (-6) = -2$ . Multiply the rise: $\frac{1}{3} \cdot (-9) = -3$ . Add to A: $(4 + (-2),\ 7 + (-3)) = (2, 4)$ . So $P = (2, 4)$ .
Total parts: $3 + 5 = 8$ . Fraction: $k = \frac{3}{8}$ . Length: $16 - (-8) = 24$ . Multiply: $\frac{3}{8} \cdot 24 = 9$ . Add to the start: $-8 + 9 = 1$ . The point is at $1$ .
Total parts: $1 + 3 = 4$ . Fraction: $k = \frac{1}{4}$ . Run: $9 - 1 = 8$ . Rise: $1 - (-3) = 4$ . Multiply the run: $\frac{1}{4} \cdot 8 = 2$ . Multiply the rise: $\frac{1}{4} \cdot 4 = 1$ . Add to A: $(1 + 2,\ -3 + 1) = (3, -2)$ . So $P = (3, -2)$ .
Total parts: $2 + 3 = 5$ . Fraction: $k = \frac{2}{5}$ . Run: $4 - (-1) = 5$ . Rise: $4 - 2 = 2$ . Multiply the run: $\frac{2}{5} \cdot 5 = 2$ . Multiply the rise: $\frac{2}{5} \cdot 2 = \frac{4}{5}$ . Add the x-values: $-1 + 2 = 1$ . Add the y-values: $\frac{10}{5} + \frac{4}{5} = \frac{14}{5}$ . So $P = \left(1, \frac{14}{5}\right)$ .
Total parts: $1 + 1 = 2$ . Fraction: $k = \frac{1}{2}$ . Run: $2 - (-6) = 8$ . Rise: $-8 - 4 = -12$ . Multiply the run: $\frac{1}{2} \cdot 8 = 4$ . Multiply the rise: $\frac{1}{2} \cdot (-12) = -6$ . Add to A: $(-6 + 4,\ 4 + (-6)) = (-2, -2)$ . So $P = (-2, -2)$ . This point is the midpoint of $\overline{AB}$ .
The fraction is: $k = \frac{1}{1 + 3} = \frac{1}{4}$ . The move from A to P is: $(2 - 0,\ 1 - 0) = (2, 1)$ . That move is one quarter of the run and rise, so multiply it by 4: $(4 \cdot 2,\ 4 \cdot 1) = (8, 4)$ . Add to A: $B = (0 + 8,\ 0 + 4) = (8, 4)$ . So $B = (8, 4)$ . Check: $\frac{1}{4} \cdot 8 = 2$ and $\frac{1}{4} \cdot 4 = 1$ .
Fraction: $k = \frac{1}{4}$ . Run: $9 - (-3) = 12$ . Rise: $12 - (-4) = 16$ . Multiply the run: $\frac{1}{4} \cdot 12 = 3$ . Multiply the rise: $\frac{1}{4} \cdot 16 = 4$ . Add to A: $(-3 + 3,\ -4 + 4) = (0, 0)$ . The gas station is at $(0, 0)$ .

FAQ

What does it mean to partition a line segment?

It means finding the point that splits the segment into two parts with a given ratio, like $2:3$ .

Why is the fraction a over a + b and not a over b?

The ratio $a:b$ compares the two parts to each other. To find how far along the whole segment to go, compare the first part to all the parts: $\frac{a}{a + b}$ .

Does the order of the points matter?

Yes. Splitting from $A$ to $B$ in the ratio $1:3$ lands near $A$ . Splitting from $B$ to $A$ lands near $B$ .

How is this related to the midpoint formula?

The midpoint is the partition with ratio $1:1$ . Then $k = \frac{1}{2}$ , and the formula gives the average of the coordinates.

Summary

To split $\overline{AB}$ in the ratio $a:b$ from $A$ , use $k = \frac{a}{a + b}$ .
Move $k$ of the run and $k$ of the rise from $A$ : $P = \big(x_1 + k(x_2 - x_1),\ y_1 + k(y_2 - y_1)\big)$ .
The order of the points matters. Always start at the first point named.
The ratio $1:1$ gives the midpoint.
Check that the point sits on the segment, on the correct side.

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