LSSCP - Partitioning a Line Segment Lesson

Partitioning a Line Segment

A directed line segment is a segment between two points A and B with a specified direction, from A to B or from B to A. To partition means to separate, or divide. To partition a directed line segment is to divide it into two segments with a given ratio. We can partition a line segment in a given ratio by following the steps below.

$Image with Steps for Partitioning a Line Segment: 1. Label your points (x1,y1) and (x2, y2). Note: Since it is a directed segment, order does matter. Be sure to list the starting point as (x1, y1, and the ending point as (x2, y2). 2. Convert the given ratio to a percent. Keep it as a fraction. a:b-->[a/(a+b)]. This fraction is your percent ratio. 3. Find the rise and run for the segment. (Remember, order DOES matter!) rise = y2-y1 run = x2-x1 4. To find the partitioning point, plug your information from steps #1-3 into the following formula. x-coordinate: x1+(run)(a/(a+b)) y-coordinate: y1+(rise)(a/(a+b)) So the point is: [x1+(run)(a/(a+b)), y1+(rise)(a/(a+b))]$

Let's try an example!

Example 1: Find the coordinate of point P that lies along the directed line segment from A(3, 4) to B(6, 10) and partitions the segment in the ratio of 3 to 2.

Directed Line segment with A plotted at (3,4) and B being at (6, 10) and P being a plot on the line segment

Step 1: (x₁,y₁) is (3, 4) and (x₂, y₂) is (6, 10)

Step 2: Convert ratio of 3:2 to $LaTeX: \frac{3}{3+2}=\frac{3}{5}$ $\frac{3}{3+2}=\frac{3}{5}$ equation image indicator

Line Segment AB being plotted on a graph
A(3, 4)
B(6, 10)

Step 3:

rise = 10 - 4 = 6

run = 6 - 3 = 3 (notice on this example, we could have just counted since we had the graph)

Step 4:

x-coordinate = $LaTeX: x_1+\left(run\right)\left(\frac{a}{a+b}\right)=3+\left(3\right)\left(\frac{3}{3+2}\right)=3+\frac{9}{5}=4.8$ $x_1+\left(run\right)\left(\frac{a}{a+b}\right)=3+\left(3\right)\left(\frac{3}{3+2}\right)=3+\frac{9}{5}=4.8$ equation image indicator

y-coordinate = $LaTeX: y_1+\left(rise\right)\left(\frac{a}{a+b}\right)=4+\left(6\right)\left(\frac{3}{3+2}\right)=4+\frac{18}{5}=7.6$ $y_1+\left(rise\right)\left(\frac{a}{a+b}\right)=4+\left(6\right)\left(\frac{3}{3+2}\right)=4+\frac{18}{5}=7.6$ equation image indicator

So the coordinates of point P are (4.8, 7.6). This is the point that partitions the segment in a 3:2 ratio.

Let's try another one...

Example 2: Find the coordinates of point P along the directed line segment from A(1,3) to B(8, 4) so that AP to PB gives a ratio of 4:1.

Step 1: (x₁,y₁) is (1, 3) and (x₂, y₂) is (8, 4).

Step 2: Convert ratio of 4:1 to $LaTeX: \frac{4}{4+1}=\frac{4}{5}$ $\frac{4}{4+1}=\frac{4}{5}$

Step 3:

rise = 4 - 3 = 1

run = 8 - 1 = 7

Step 4:

x-coordinate = $LaTeX: x_1+\left(run\right)\left(\frac{a}{a+b}\right)=1+\left(7\right)\left(\frac{4}{4+1}\right)=1+\frac{28}{5}=6.6$ $x_1+\left(run\right)\left(\frac{a}{a+b}\right)=1+\left(7\right)\left(\frac{4}{4+1}\right)=1+\frac{28}{5}=6.6$

y-coordinate = $LaTeX: y_1+\left(rise\right)\left(\frac{a}{a+b}\right)=3+\left(1\right)\left(\frac{4}{4+1}\right)=3+\frac{4}{5}=3.8$ $y_1+\left(rise\right)\left(\frac{a}{a+b}\right)=3+\left(1\right)\left(\frac{4}{4+1}\right)=3+\frac{4}{5}=3.8$

So the coordinates of point P are (6.6, 3.8). This is the point that partitions the segment in a 4:1 ratio.

$Image with: Note: If the ratio is already in rational form, you do not have to convert it! Example: You may see *Find the point that is (2/5) of teh way between A(2,6) and point B(-1,4) instead of *Find the point that divides the segment between points A(2,6) and point B(-1,4) in a 2:3 ratio but they are the SAME problem! Don't let it fool you. It just saves you the step of converting the original ratio to a fraction!$

Example 3: Find the coordinates of point P that is ¼ of the way along the directed line segment from A(2, -2) to B(3, 4).

Step 1: $LaTeX: \left(x_1,\:y_1\right)is\left(2,\:-2\right)\:and\:\left(x_2,\:y_2\right)is\:\left(3,\:4\right)$ $\left(x_1,\:y_1\right)is\left(2,\:-2\right)\:and\:\left(x_2,\:y_2\right)is\:\left(3,\:4\right)$

Step 2: We don't have to do step 2 since our ratio is already in fraction form: 1/4!

Step 3:

rise = 4 - (-2) = 4 + 2 = 6

run = 3 - 2 = 1

Step 4:

x - coordinate = $LaTeX: x_1+\left(run\right)\left(\frac{a}{a+b}\right)=2+\left(1\right)\left(\frac{1}{4}\right)=2+\frac{1}{4}=2\frac{1}{4}or\:2.25$ $x_1+\left(run\right)\left(\frac{a}{a+b}\right)=2+\left(1\right)\left(\frac{1}{4}\right)=2+\frac{1}{4}=2\frac{1}{4}or\:2.25$ equation image indicator

y - coordinate = $LaTeX: y_1+\left(rise\right)\left(\frac{a}{a+b}\right)=-2+\left(6\right)\left(\frac{1}{4}\right)=-2+\frac{6}{4}=-\frac{1}{2}\:or\:-0.5$ $y_1+\left(rise\right)\left(\frac{a}{a+b}\right)=-2+\left(6\right)\left(\frac{1}{4}\right)=-2+\frac{6}{4}=-\frac{1}{2}\:or\:-0.5$ equation image indicator

So, the coordinates of point P are $LaTeX: \left(2\frac{1}{4},\:-\frac{1}{2}\right)$ $\left(2\frac{1}{4},\:-\frac{1}{2}\right)$ . This is the point that is 1/4 of the way from A to B, partitioning the segment in a 1:3 ratio!

Now that you know how to partition a directed line segment, try the practice problems below!

IMAGES CREATED BY GAVS