SLO 2.5 : Directed Line Segments - Direct Instruction

SLO 2.5 : Directed Line Segments -

Direct Instruction

By Christopher Morley

​SLO 2.5 - Partitioning A Line Segment

How do you find the point on a segment that partitions the segment by a given ratio?

Recall that the slope of a straight line graphed in the coordinate plane is the ratio of the rise to the run. In the figure to the right, what is the slope?

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In this lesson, we will be exploring the concept of the directed line segment.  This means the segment has direction associated with it, usually specified by moving from the first named endpoint (A in the example to the right) to the other endpoint (B).

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At a point (say P) that partitions the line segment into two, a ratio expresses the relationship of the relative size of the two parts. You need to know the rise and the run to determine where the point P sits on the directed line segment. As always you can either count the boxes on the coordinate plane or use the coordinates themselves:

​RISE: (y₂ - y₁)

RUN: (x₂ - x₁) 

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Steps for finding "P"

​3) Dilate the Rise and Run

Multiply the rise and the run by the new “part-to-whole” ratio. This dilates the sides of the slope triangle to form a similar smaller triangle that will have the first point (A) and the partition point (P) as its vertices on the original line segment.

Record the product of the rise times the "part-to-whole" ratio and the product of the run times the same ratio.

​1) Part:Part -> Part:Whole

Use the given ratio to determine the total number of pieces the line segment will be broken into.  Unless otherwise stated, the ratio given in the problem is usually the “part-to-part” ratio.

Add the two numbers to get the total number of pieces in the whole line segment.​

​2) Rise and Run

Draw a slope triangle or use slope formula. Find the rise, then find the run.

You'll save confusion if you use the coordinates of the first point as (x₁,y₁) if you use the slope formula​

​4) Add these products to the first point

Add the result of the run times the part-to-whole ratio to A's (the first point) x-coordinate. Do the same for the rise times the part to whole to A's y-coordinate.

These sums will be the x, y coordinates of P, the partition point. Drawing the dilated slope triangle will also show P's position.

​Dilate the Rise and Run

Multiply the rise and the run by the new “part-to-whole” ratio. This dilates the sides of the slope triangle to form a similar smaller triangle that will have the first point (A) and the partition point (P) as its vertices on the original line segment.

Image of rise is 6 x 3/5 = 18/5 = 3 3/5 or 3.6

Image of run is 3 x 3/5 = 9/5 = 1 4/5 or 1.8

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​Rise and Run

Draw a slope triangle or use slope formula. Find its rise, then find its run.

(y₂ - y₁) = (10 - 4) = 6 <- rise

(x₂ - x₁) = (6 - 3) = 3 <- run​

​Add these products to the first point

Add the result of the (run * the part-to-whole ratio) to the first point's (A's) x-coordinate. Do the same for the (rise * the part-to-whole ratio) to A's y-coordinate.

A (3,4) + [1.8, 3.6] = P (4.8, 7.6)

Example: Given A(3, 4) and B (6, 10).

Find the point P on AB such that 3·AP= 2·PB, or the ratio of |AP| to |PB| is 3/2

Multiple Choice

You can see the orange dilated slope triangle is $$\frac{3}{5}$$  the size of the original red slope triangle. When you add the lengths of the reduced rise and run to point A's, what are the coordinates of point P?

• (6, 10)

• (3, 4)

• (5, 8)

• (4.8, 7.6)

Multiple Choice

Given A (1, 1) and B (7, 3). Find the point P on AB such that 1·AP = 4·PB.

Step 1: Convert the part-to-part ratio to

part-to-whole

• 1 : 4

• 1 : 5

• 3 : 4

• 4 : 5

Multiple Choice

Given A (1, 1) and B (7, 3). Find the point P on AB such that 1·AP = 4·PB.

Step 2: Determine the rise and the run of the slope

• $$\frac{\left(3\ -\ 1\right)}{\left(7\ -\ 1\right)}$$  

• $$\frac{\left(1\ -\ 3\right)}{\left(1\ -\ 7\right)}$$  

• $$\frac{\left(7\ -\ 3\right)}{\left(1\ -\ 1\right)}$$  

• $$\frac{\left(7\ -\ 1\right)}{\left(3\ -\ 1\right)}$$  

Multiple Select

Given A (1, 1) and B (7, 3). Find the point P on AB such that 1·AP = 4·PB.

Step 3: Dilate the slope triangle by the part-to-whole ratio.

(Choose all correct answers).

• $$2\ \times\frac{1}{5}$$  

  RISE 

• $$2\ \cdot\ \frac{1}{4}$$   

  RISE

• $$6\ \cdot\ \frac{1}{5}$$   

  RUN

• $$6\ \cdot\ \frac{1}{4}$$  

  RUN 

Multiple Choice

Given A (1, 1) and B (7, 3). Find the point P on AB such that 1·AP = 4·PB.

Step 4: Add the dilated rise to the y-coordinate of the starting point. Add the dilated run to the x-coordinate of the starting point.

(These will be the coordinates of point P).

• $$P=\left(1,1\right)$$  

• $$P=\left(1,5\right)$$  

• $$P=\left(1.2,0.4\right)$$

• $$P=\left(2.2,1.4\right)$$