y5

0x

-5

-5

5Rise = 1 - (-3) = 4

Run = 4 - (-4) = 8

A(-4, -3)

B(4, 1)

A(3, 4)

B(6, 10)

P

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Name   Class    Date    3-5

Partitioning a Segment

Find the coordinates of the point P that lies along the directed line segmentfrom A(3, 4) to B(6, 10) and partitions the segment in the ratio 3 to 2.

A Convert the ratio to a percent.

Point P is 3 _____ 3 + 2 = 3 __ 5 of the distance from A to B.

This is % of the distance from A to B.

B Find the rise and run for ___

AB.

Rise = 10 - 4 = 6 Run =

C The slope of ___

AP must be the same as the slope of ___

AB.

So, to find the coordinates of P, add % of the run to the x-coordinate of

A and add % of the rise to the y-coordinate of A.

x-coordinate of P = 3 + · 3 =

y-coordinate of P = 4 + · =

So, the coordinates of P are .

E X A M P L E1

Slopes of LinesExtension: Using Slope to Partition SegmentsEssential question:Howdoyoufindthepointonadirectedlinesegmentthatpartitionsthesegmentinagivenratio?

Recall that the slope of a straight line in a coordinate plane is the ratio of the rise to the run.

In the figure, the slope of ___

AB is rise ___ run = 4 __ 8 = 1 __ 2 .

In the next several lessons, you will see how to use slope to solve geometry problems and to prove geometry theorems.

The following example also uses the idea of a directed linesegment. This means the line segment has a direction associated with it, usually specified by moving from one endpoint to the other.

G-GPE.2.6

Chapter 3 113 Lesson 5

y5

50

x

-5

-5

Bedford

Ashton

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p r a c t i c e

  1.  Find the coordinates of the point P that lies along the directed segment from C(-3, -2) to D(6, 1) and partitions the segment in the ratio 2 to 1.

  3.  Find the coordinates of the point P that lies along the directed segment from J(-2, 5) to K(2, -3) and partitions the segment in the ratio 4 to 1.

  2.  Find the coordinates of the point P that lies along the directed segment from R(-3, -4) to S(5, 0) and partitions the segment in the ratio 2 to 3.

  4.  Find the coordinates of the point P that lies along the directed segment from M(5, -2) to N(-5, 3) and partitions the segment in the ratio 1 to 3.

  5.  The map shows a straight highway between two towns. Highway planners want to build two new rest stops between the towns so that the two rest stops divide the highway into three equal parts. Find the coordinates of the points at which the rest stops should be built.

  6.  ‹

__ › RS passes through R(-3, 1) and S(4, 3). Find a point P on

‹

__ › RS such

that the ratio of RP to SP is 5 to 4. Is there more than one possibility? Explain.

REFLECT

1a.  Explain how you can check that the slope of ___

AP equals the slope of ___

AB.

1b.  Explain how you can use the distance formula to check that P partitions ___

AB in the ratio 3 to 2.

 

Chapter 3 114 Lesson 5

Name ________________________________________ Date __________________ Class __________________

Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.

Holt McDougal Geometry

Practice Slopes of Lines

Use the slope formula to determine the slope of each line.

1. AB _______________________ 2. CD _______________________

3. EF _______________________ 4. GH _______________________

5. Find the coordinates of the point P that lies along the directed line segment from A(3, 1) to B(6, 7) and partitions the segment in the ratio 2 to 1.

_______________________________

7. Find the coordinates of the point P that lies along the directed line segment from E(−5, 5) to F(−2, −2) and partitions the segment in the ratio 1 to 1.5.

_______________________________

6. Find the coordinates of the point P that lies along the directed line segment from C(−3, −2) to D(5, 2) and partitions the segment in the ratio 1 to 4.

_______________________________

8. Find the coordinates of the point P that lies along the directed line segment from G(1, 1) to H(8, 1) and partitions the segment in the ratio 1 to 3.

_______________________________

Chapter

2

Sec2:115

H-CS10_G_MNLESE882000_C01.indd 115 6/26/12 10:05:09 PM

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3-5Name   Class    Date   

Additional Practice

Chapter 3 115 Lesson 5

Name ________________________________________ Date __________________ Class __________________

Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.

Holt McDougal Geometry

Problem Solving Slopes of Lines

1. The map shows a highway between Acton and Beauville and its intersection with Highway 10. A power station needs to be built along the highway between Acton and Beauville. The station must to be located so that the ratio of its distance to Acton to its distance to Beauville is 2 to 3, but the builders also want to locate it south of Highway 10. Can this be achieved? Explain why or why not.

_________________________________________________________________________________________

_________________________________________________________________________________________

_________________________________________________________________________________________

2. The segment represented by endpoints C(1, 1) and D(7, 3) is not directed. What two points would partition the segment such that the ratio of the lengths is 1 to 2?

_________________________________________________________________________________________

Select the best answer. 3. Which point partitions the directed

segment from A(−3, −2) to B(5, −1) in a ratio of 2 to 3?

A 1 21 , 15 5

æ öç ÷-ç ÷è ø

B 1 3, 15 5

æ öç ÷-ç ÷è ø

C 1 22 , 13 3

æ öç ÷-ç ÷è ø

D 1 1, 13 3

æ öç ÷- -ç ÷è ø

4. The directed line segment from C(−2, −1) to D(4, 2) is partitioned by P(2, 1). What is the ratio of CP to DP? F 1:1 H 1:2 G 2:1 J 2:3

5. A directed line segment from E(−3, 3.6) to F(5.3, 3.6) is partitioned by point P in the ratio 2:7. What is the y-coordinate of P? A 0.3 B 1.15 C 1.35 D 3.6

Chapter

2

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Problem Solving

Chapter 3 116 Lesson 5