1.1 Line Segments, Distance and Midpoint 1.1d – Partitioning a Segment

CC Standard

G-GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

At the end of this lesson, you should be able to answer the following

question:

How do you find the point on a directed line segment that partitions the segment in a given ratio?

Point P divides in the ratio 3 to 1. 1. What does this mean?

AB

AP

= -1= -7

Slope = m = =

-1

-7

1

7

m = -4

5

Slope Formula

The slope of a line through the points (x1, y1) and (x2, y2) is as follows:

yy22 –– yy11 xx22 –– xx11

m =

Ex.

Find the slope of the line that passes through (–2, –3) and (4, 6).

Let (x1, y1) be (–2, –3) and (x2, y2) be (4, 6).

6 – (–3)4 – (–2)

Substitute 6 for y2, –3 for y1, 4 for x2, and –2 for x1.

=y2 – y1

x2 – x1

96= 3

2=

The slope of the line that passes through (–

2, –3) and (4, 6) is . 32

*** Always reduce your fractions****

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:Tells the direction in which from which point to start and end. In this case, from Point A to Point B So point A must be labeled A and B

What does that tell you about the distance AP and PB in relation to AB?

Example 1:

),( 11 yx ),( 22 yx

Partitioning a segment 1. Label your points and

Note: since it is a directed segment, order does matter.

),( 11 yx ),( 22 yx

3. Find the rise and run for the segment (order does matter)

Rise: Run: 12 xx 12 yy

4. To find the partitioning point:

x – coordinate: x1 -coordinate + run(% in fraction form)

y - coordinate: y1 -coordinate + rise(% in fraction form)

2. Convert the ratio into a percent (keep as a fraction) a:b

Percent ratio (%)

ba

a

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.

2. Convert the ratio into a percent

3. Find the rise and run for AB

4. The slope of AP must be the same as AB. So, to find the coordinates of P, add 60% of the run to the x-coordinate of A and add 60% of the rise to the y-coordinate of A

How can you use the distance formula to check that P partitions in the ratio of 3 to 2?

AB

1. Label your points. (note: Since it is a directed segment, order does matter, so A(x1, y1) and

B(x2, y2) ).

Example 2:

Find the coordinates of the point P that lies along the directed segment from A(1, 1) to B(7, 3) and partitions the segment in the ratio of 1 to 4

Example 3: Find the coordinates of point P that lies along the directed line segment from M to N and partitions the segment in the ratio of 3 to 2

Essential question:

• How do you find the point on a directed line segment that partitions the segment in a given ratio?

• Class assignment:

Work sheet 1.1d – Partitioning a segment