Partitioning the Segment­ 1st Block 12­3­12
Essential Question
How do I find a point on a line
segment that separates the
segments into a given ratio?
A directed line segment
is a line segment from one
point to another point in the
coordinate plane.
Partitioning the Segment­ 1st Block 12­3­12
The components of directed line segment
AB shown below are x2 ­ x1, y2 ­ y1 .
The components describe the direction
and length of the directed line segment.
Example 1:
The components of the vector from P to Q (the
directed line segment PQ) are 9 ­ 5, 6 ­ 4 = 4, 2 .
They tell you that a "route" from P to Q is 4 units
right and 2 units up. Note that the components are
used in the slope:
6­4 = 2 = 1
9­5
4
2
Partitioning the Segment­ 1st Block 12­3­12
To find a point that partitions a directed line segment in a given ratio, use the following steps:
Step 1: Look at the ratio given. As an example, a
ratio can be expressed as 2 to 3, 2:3, or 2/3.
In order to divide the segment in the ratio of
2 to 3, think of dividing, or partitioning, the
segment into 2 + 3, or 5, congruent pieces.
Therefore, point R is 2/5 of the way FROM the
first endpoint to the second endpoint. To find
the correct fraction in any problem, find:
= 2 = 2
numerator
5
numerator + denominator 2 + 3
Step 2: Find the components of the directed line
segment using x2 ­ x1, y2 ­ y1 .
P(5, 4) and Q(9, 6) so 9 ­ 5, 6 ­ 4 is 4, 2 .
Partitioning the Segment­ 1st Block 12­3­12
Step 3: Multiply the fraction found in step 1 by
the components found in step 2.
2
5
4 = 8 or 1.6 and
5
2
5
2 = 4 or 0.8
5
Step 4: Add the answer from step 3 to the first
endpoint. This will give the coordinates of the
partitioning point R.
P(5, 4) ⇒ R(5 + 1.6, 4 + 0.8) ⇒ R(6.6, 4.8)
Partitioning the Segment­ 1st Block Example 2:
Find the coordinates of
point P along the directed
line segment AB so that the
ratio of AP to PB is 3 to 2.
Example 3:
Given the points A(­1, 2) and B(7, 8), find the
coordinates of point P on directed line segment
AB that partitions AB in the ratio 1/3.
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Partitioning the Segment­ 1st Block Example 4:
Given the points A(­4, 2) and B(2, 8), find the
coordinates of point P on directed line segment
AB that partitions AB in the ratio 3/4.
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