1-5 Measuring Segments

1-5

Measuring Segments

What You'll Learn

? To find the lengths of

segments

. . . And Why

To find distance using a highway number line, as in Exercise 23

Check Skills You'll Need

GO for Help Skills Handbook pages 757 and 758

Simplify each absolute value expression.

1. u-6u 6

2. u3.5u 3.5

4. u-4 - 2u 6

5. u-2 - (-4)u 2

3. u7 - 10u 3 6. u-3 + 12u 9

x2 Algebra Solve each equation.

7. x + 2x - 6 = 6 4

8. 3x + 9 + 5x = 81 9

9.

w

-

2

=

-4

+

7w

1 3

New Vocabulary ? coordinate ? congruent segments ? midpoint

1 Finding Segment Lengths

The distance between points C and D on the ruler is 3. You can use the Ruler Postulate to find the distance between points on a number line.

C

D

2345678

Key Concepts

Postulate 1-5 Ruler Postulate

The points of a line can be put into one-to-one correspondence with the real numbers so that the distance between any two points is the absolute value of the difference of the corresponding numbers.

Vocabulary Tip

The congruence symbol (>) shows that two figures are equal (=) in size and similar (,) in shape.

the length of AB AB = ua - bu

A

B

a

b

coordinate of A coordinate of B

Two segments with the same length are congruent (>) segments. In other words, if AB = CD, then AB > CD. You can use these statements interchangeably.

2 cm

A

B

A

2 cm

C

D

C

B

AB = CD

AB > CD

D

As illustrated above, segments can be marked alike to show they are congruent.

1-5

1. Plan

Objectives

1 To find the lengths of segments

Examples

1 Comparing Segment Lengths 2 Using the Segment Addition

Postulate 3 Using the Midpoint

Math Background

A one-to-one correspondence, as used in the Ruler Postulate, is one way to show that two sets are equivalent when their elements cannot be counted. The Ruler Postulate is an abstract description of how the measurement tool works. Although he did not list it as a postulate, Euclid implicitly used the Segment Addition Postulate in his proofs.

More Math Background: p. 2D

Lesson Planning and Resources

See p. 2E for a list of the resources that support this lesson.

PowerPoint

Bell Ringer Practice

Check Skills You'll Need For intervention, direct students to: Skills Handbook, pp. 757, 758

Lesson 1-5 Measuring Segments 31

Special Needs L1 Have students use a ruler to demonstrate the Segment Addition Postulate.

Below Level L2 Review the definition of absolute value. Ask: Why do you think that absolute value is used to express the distance between two points? Distance is a positive number.

learning style: tactile

learning style: verbal

31

2. Teach

Guided Instruction

Error Prevention!

The statements AB CD and AB = CD are interchangeable. However, point out that AB = CD is never correct. The equal sign only compares numbers, never geometric figures.

1 EXAMPLE Alternative Method

Students may count along the number line to find the lengths. This will help them better understand the Ruler Postulate.

PowerPoint

Additional Examples

1 Find which two of the

segments XY , ZY , and ZW are

congruent.

X

Y

Z

W

?6?5?4?3?2?1 0 1 2 3 4 5 6

XY ZW and XZ YW

2 If AB = 25, find the value of x. Then find AN and NB.

2x ? 6

x 7

A

N

B

x 8; AN 10, NB 15

3 M is the midpoint of RT. Find RM, MT, and RT.

5x + 9

8x 36

R

M

T

RM 84, MT 84, RT 168

Resources

? Daily Notetaking Guide 1-5 L3

? Daily Notetaking Guide 1-5--

Adapted Instruction

L1

Closure

Explain how Segment Addition Postulate can help you measure segments. Sample: Segment Add. Post. states that the sum of all the parts of a segment equals the whole segment.

32

1 EXAMPLE Comparing Segment Lengths

Find AB and BC. Are AB and BC congruent?

A

B

C

D

E

8 7 6 5 4 3 2 1 0 1 2 3

AB = u-8 - (-5)u = u-3u = 3 BC = u-5 - (-2)u = u-3u = 3 AB = BC, so AB > BC.

Quick Check

1 a. Compare CD and DE. Are the segments congruent? CD 5 DE ; yes b. Critical Thinking To find AB in Example 1, suppose you subtract -8 from -5. Do you get the same result? Why? yes; ?25 2 (28)... 5 ?3... 5 3

Key Concepts

Examine the lengths of AB and BC in Example 1. Notice that AB + BC = 6. Notice that AC = 6. This suggests the following postulate.

Postulate 1-6 Segment Addition Postulate

If three points A, B, and C are collinear and

A

B is between A and C, then AB + BC = AC.

B C

2 EXAMPLE Using the Segment Addition Postulate

Algebra If DT = 60, find the value of x. Then find DS and ST.

2x 8 3x 12

D

S

T

Quick Check

DS + ST = DT (2x - 8) + (3x - 12) = 60

5x - 20 = 60 5x = 80 x = 16

DS = 2x - 8 = 2(16) - 8 = 24 ST = 3x - 12 = 3(16) - 12 = 36

Segment Addition Postulate Substitute. Simplify. Add 20 to each side. Divide each side by 5.

Substitute 16 for x.

2 EG = 100. Find the value of x. 4x 20 2x 30

Then find EF and FG. 15; EF 40, FG 60

E

F

G

A midpoint of a segment is a point that divides the segment into two congruent segments. A midpoint, or any line, ray, or other segment through a midpoint, is said to bisect the segment.

A

B

C

AB BC

32 Chapter 1 Tools of Geometry

Advanced Learners L4 Have students investigate "one-to-one correspondence" as it applies to algebraic functions.

learning style: verbal

English Language Learners ELL Review how congruent segments have equal lengths. Then write AB on the board and state that this denotes the length of the corresponding segment, AB.

learning style: visual

3 EXAMPLE Using the Midpoint

Algebra C is the midpoint of AB. Find AC, CB, and AB.

2x + 1

3x - 4

A

C

B

AC = CB 2x + 1 = 3x - 4 2x + 5 = 3x

5=x AC = 2x + 1 = 2(5) + 1 = 11 CB = 3x - 4 = 3(5) - 4 = 11

Definition of midpoint Substitute. Add 4 to each side. Subtract 2x from each side.

Substitute 5 for x.

AC and CB are both 11, which is half of 22, the length of AB.

Quick Check 3 Z is the midpoint of XY, and XY = 27. Find XZ. 13.5

.

EXERCISES

For more exercises, see Extra Skill, Word Problem, and Proof Practice.

Practice and Problem Solving

A Practice by Example

Find the length of each segment. Tell whether the segments are congruent.

GO

for Help

Example 1 (page 32)

9, 9; 1. AC and BD yes 2. BD and CE 9, 6; no A B

CD E

3. AD and BE 11, 13; no

4. BC and CE 7, 6; no 8 6

13

7

On a number line, the coordinates of X, Y, Z, and W are ?7, ?3, 1, and 5, respectively. Find the lengths of the two segments and tell whether they are congruent.

Example 2 (page 32)

5. XY and ZW XY ZW 4; yes

6. ZX and WY ZX WY 8; yes

Use the figure at the right for Exercises 8?11.

8. If RS = 15 and ST = 9, then RT = . 24 R

9. If ST = 15 and RT = 40, then RS = . 25

7. YZ and XW YZ 4; XW 12; no

S

T

x2 10. a. Algebra If RS = 3x + 1, ST = 2x - 2, and RT = 64, find the value of x.

b. Find RS and ST. 10a. 13

10b. RS 40, ST 24

x2 11. a. Algebra If RS = 8y + 4, ST = 4y + 8, and RT = 15y - 9, find the value of y. 7

b. Find RS, ST, and RT. RS 60, ST 36, RT 96

Example 3 x2 12. Algebra A is the midpoint of XY.

(page 33)

a Find XA. 9

b. Find AY and XY. 9; 18

3x 5x - 6

X

A

Y

x2 Algebra In Exercises 13 ?15, use the figure and find PT.

13. PT = 5x + 3 and TQ = 7x - 9 33

P

T

Q

14. PT = 4x - 6 and TQ = 3x + 4 34

15. PT = 7x - 24 and TQ = 6x - 2 130

Lesson 1-5 Measuring Segments 33

3. Practice

Assignment Guide

1 A B 1-35 C Challenge

36-38

Test Prep Mixed Review

39-44 45-55

Homework Quick Check

To check students' understanding of key skills and concepts, go over Exercises 6, 12, 20, 32, 34.

Visual Learners Exercises 5?7 Have students

draw a number line to represent each problem.

Error Prevention!

Exercises 13?15 Students may think they are finished after they solve for x. Remind them to read the directions carefully.

GPS Guided Problem Solving

L3

Enrichment

L4

Reteaching

L2

Adapted Practice

L1

PraNcamte ice

Class

Date

L3

Practice 1-5

Construct each figure as directed.

1. Construct AB congruent to XY. Check your work with a ruler.

2. Construct the perpendicular bisector of XY.

X

3. Construct a triangle whose sides are all the same length as XY.

4. Construct the angle bisector of &Z.

Basic Constructions Y

Z

Check your work with a protractor. 5. a. Construct a 90? angle. b. Construct a 45? angle. 6. Construct AB so that AB = MN + OP. 7. Construct KL so that KL = OP - MN.

8. Construct &A so that m&A = m&1 + m&2. 9. Construct &B so that m&B = m&1 - m&2. 10. Construct &C so that m&C = 2m&2.

11. Construct the angle bisector of &X.

12. Construct &W so that m&W = 2m&X.

13.

Construct

&Z

so

that

m&Z

=

1 2

m&X.

M

NO

P

1

2

X

? Pearson Education, Inc. All rights reserved.

Write true or false.

14. AB XY

6 cm

6 cm

Y

A

BX

15. m&1 = 40 60 1

16. If m&A = 80, then &A is obtuse.

17. The perpendicular bisector of a line segment creates four 90? angles.

18. If m&1 = 45 and m&2 = m&1, then m&1 + m&2 = 90.

19.

For

a

given

& A,

1 2

?

m& A

=

2

?

m& A.

20. If angles 3 and 4 are complementary and m&3 = m&4, then m&4 = 45.

33

Exercises 17?19 Each exercise asks for a coordinate. Students may think they are asked to name a point on the diagram by a letter, instead of giving a number.

Exercise 31 Solve this exercise together as a class, as it highlights that the Segment Addition Postulate does not apply to overlapping segments.

B Apply Your Skills

GO nline

Homework Help

Visit: Web Code: aue-0105

Use the figure at the right for Exercises 16?19.

16. Find the midpoint of AB. Q

W

A SQ

B

17. What is the coordinate of the

8

024

8

midpoint of QB? 6

18. What is the coordinate of the midpoint of WA? ?4

19. What is the coordinate of the midpoint of the segment formed by the two points you found in Exercises 17 and 18? 1

Suppose the coordinate of A is 0, AR 5, and AT 7. What are the possible coordinates of the midpoint of the given segment?

20. AR ?2.5, 2.5

21. AT ?3.5, 3.5

22. RT ?6, ?1, 1, 6

23. Mileage Exit numbers on some highways give mileage from one edge of the state. The highway and its exit numbers resemble a number line. You can find distance between exits in the same way that you find distance on a number line.

a. Use the map. How many miles is it from Wildorado to Shamrock? 114 miles b. Which town is 28 miles from Jericho? Conway

Visualization Without using your ruler, sketch a segment with the given length. Then use your ruler to see how well you did. 24?28. Check students' work.

24. 3 cm

25. 3 in.

26. 6 in.

27. 10 cm

28. 65 mm

Vocabulary Tip

For Exercise 30, you are to decide whether the distance BD is less than the distance CD.

In Exercises 29?32, describe the statement as true or false. Explain. 29?32. See margin.

AB 8 6

CD E

13

7

29. AB > CD

30. BD , CD

Exercises 29?33

31. AC + BD = AD

32. AC + CD = AD

33. Suppose EG = 5. Find the possible coordinate(s) of point G. 2, 12

34. y 15; AC 24, DC 12

35. ED 10, DB 10, EB 20

x2 Algebra Use the diagram at the right for Exercises 34 and 35.

GPS 34. If AD = 12 and AC = 4y - 36, find

A

the value of y. Then find AC and DC.

35. If ED = x + 4 and DB = 3x - 8, find ED, DB, and EB.

D E

B C

C Challenge

36. C is the midpoint of AB, D is the midpoint of AC, E is the midpoint of AD, F is the midpoint of ED, G is the midpoint of EF, and H is the midpoint of DB. If DC = 16, find GH. 30

37. a. Write an algebraic expression that represents GK. 5x

b. If GK = 30, find GH and JK. 9, 15

4x 3

2x 3 x

G HJ

K

34 Chapter 1 Tools of Geometry

24. 60; acute

25. 90; right

26. 135; obtuse

27. 34

34

28. 70

38. Art Project You want to cut pieces of ribbon for an art project. Each piece must be 634 inches long. For measuring, you have only the old, broken ruler shown at the right. a. What marks on the ruler would you use to measure the ribbon? b. Writing You also need 5-in. pieces of string. Describe two ways you can use the broken ruler to measure 5 inches. a?b. See back of book.

5 6 7 8 9 10 11

Test Prep

Multiple Choice

39. If KC = 31, what is KN? C

A. 43

B. 62

C. 74

D. 82

40. If KN = 29, what is CN? F

F. 13

G. 14.5

2x + 10 4x + 1

K

C

N

Exercises 39?41

H. 15.5

J. 16

41. If C is the midpoint of KN, what is KC? D

A. 4.5

B. 9

C. 18

D. 19

44. [2] 2x + 8 + 0.5(2x + 8) = 42;

x = 10 [1] correct equation with

minor error in calculation

42. On a number line, point A has coordinate ?6, and B has coordinate 2.

Which is the coordinate of point M, the midpoint of segment AB? H

F. 4

G. 0

H. ?2

J. ?3

43. Points X, Y, and Z are collinear with Y between X and Z. Which of the

following must be true? B

A. XY 5 YZ

B. XZ 2 XY 5 YZ

C. XY 1 XZ 5 YZ

D. XZ 5 XY 2 YZ

Short Response

44. Points L, M, and N are collinear with M between L and N. LM 5 2x 1 8 and MN is one half the length of LM. If LN 5 42, write and solve an equation to find x.

Mixed Review

Lesson 1-4

GO

for Help

Lesson 1-3

Complete each statement with always, sometimes, or never to make a true statement. 45. Skew lines are 9 coplanar. never 46. Skew lines 9 intersect. never 47. Opposite rays 9 form a line. always 48. Parallel planes 9 intersect. never 49. Three points are 9 coplanar. always 50. Two points are 9 collinear. always 51. The intersection of two planes is 9 a line. sometimes 52. Intersecting lines are 9 parallel. never

Lesson 1-1

Find the next two terms in each sequence.

53. 5, 10, 15, 20, c 25, 30

54. 5, 25, 125, 625, c 3125; 15,625

55. 14, 18, 22, 26, c 30, 34

4. Assess & Reteach

PowerPoint

Lesson Quiz

Use the figure below for Exercises 1?3.

X

T

Z

1. If XT = 12 and XZ = 21, then TZ = 7. 9

2. If XZ = 3x, XT = x + 3, and TZ = 13, find XZ. 24

3. Suppose that T is the midpoint

of XZ.

If XT = 2x + 11 and XZ = 5x + 8, find the value of x. 14

Alternative Assessment

Have students draw diagrams to illustrate the Segment Addition Postulate. Then have them write examples that use the postulate to find a missing measurement when two of the three measurements are known.

Test Prep

Resources For additional practice with a variety of test item formats: ? Standardized Test Prep, p. 75 ? Test-Taking Strategies, p.70 ? Test-Taking Strategies with

Transparencies

lesson quiz, , Web Code: aua-0105

Lesson 1-5 Measuring Segments 35

29. true; AB 2, CD 2 30. false; BD 9, CD 2 31. false; AC 9, BD 9,

AD 11, and 9 ? 9 u 11

32. true; AC 9, CD 2, AD 11, and 9 ? 2 11

35

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