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20 Chapter 1 Tools of Geometry

Objective o nd and compare lengths of segments

The fish isnt atzero, but you canstill find how longit is.

Measuring Segments1-3

On a freshwater fishing trip, you catch the fish below. By law, you

must release any fish between 15 and 19 in. long. You need to measureyour fish, but the front of the ruler on the boat is worn away. Can youkeep your fish? Explain how you found your answer.

In the Solve It, you measured the length of an object indirectly.

Essential Understanding You can use number operations to nd and comparethe lengths of segments.

Postulate 1-5 Ruler Postulate

Every point on a line can be paired with a real

number. Tis makes a one-to-one correspondence

between the points on the line and the real numbers.

Te real number that corresponds to a point is called

the coordinateof the point.

A B

a b

coordinate ofA coordinate of B

ate of

LessonVocabulary

coordinate distance congruent

segments midpoint segment bisector

Te Ruler Postulate allows you to measure lengths of segments

using a given unit and to nd distances between points on anumber line. Consider

*

AB)

at the right. Te

distancebetween

points Aand Bis the absolute value of the difference of their

coordinates, or u a 2 b u . Tis value is also AB, or the length of AB.

A

ABaba b

B

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Problem 1

Got It?

Problem 2

Got It?

Lesson 1-3 Measuring Segments 21

Measuring Segment Lengths

What is ST?

Te coordinate ofS

is2

4.Te coordinate of Tis 8.

ST5 u24 2 8u Definition of distance

5 u212u Subtract.

5 12 Find the absolute value.

1. What areUVand SVon the number line above?

6 4 2 0 2 4 6 8 10 12 14 16

S T U V

Ruler Postulate

What are you tryingto find?

STrepresents the lengthof ST, so you are tryingto find the distancebetween points Sand T.

If three pointsA,B, andCare collinear andBis

betweenAand C, then AB 1 BC5 AC.

AC

A B C

AB BC

Using the Segment Addition Postulate

Algebra If EG 5 59, what are EFand FG?

EF1 FG5 EG Segment Addition Postulate

(8x2 14) 1 (4x1 1) 5 59 Substitute.

12x2 13 5 59 Combine like terms.

12x5 72 Add 13 to each side.

x5 6 Divide each side by 12.

Use the value ofxto nd EFand FG.

EF 5 8x2 14 5 8(6) 2 14 5 48 2 14 5 34

FG 5 4x1 1 5 4(6) 1 1 5 24 1 1 5 25

2. In the diagram, JL 5 120. What areJK and KL?

E

8x14 4x1

F G

Use the Segment Addition Postulate towrite an equation.

Uw

EFand FG

EEG 5 59EF 5 8x2 14FG 5 4x1 1

Substitute 6 forx.

J

4x6 7x15

K L

Postulate 1-6 Segment Addition Postulate

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Problem 3

Got It?

22 Chapter 1 Tools of Geometry

When numerical expressions have the same value, you say that they are equal (5).

Similarly, if two segments have the same length, then the segments are

congruent (O) segments.

Tis means that if AB 5 CD, then AB> CD. You can also say that if AB> CD, thenAB 5 CD.

As illustrated above, you can mark segments alike to show that they are congruent.

If there is more than one set of congruent segments, you can indicate each set with

the same number of marks.

Comparing Segment Lengths

Are ACand BDcongruent?

AC 5 u22 2 5u 5 u27u 5 7

BD 5 u 3 2 10u 5 u27u 5 7

Yes. AC5 BD, so AC> BD.

3. a. Use the diagram above. Is ABcongruent to DE?

b. Reasoning o nd ACin Problem 3, suppose you subtract 22 from 5.

Do you get the same result? Why?

Te midpointof a segment is a point that divides the segment into two congruent

segments. A point, line, ray, or other segment that intersects a segment at its midpoint is

said to bisectthe segment. Tat point, line, ray, or segment is called a segment bisector.

A B

ABCD

C D

A B

C D

1 in.

1 in.

ABCD

A

6 4 2 0 2 4 6 8 10 12 14 16

B C D E

Definition of distance

Bis the midpoint

ofAC

.

is a segment

bisector ofAC

.

AA BB

CC

How do you knowif segments arecongruent?

Congruent segmentshave the same length. Sofind and compare the

lengths of ACand BD.

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Problem 4

Got It?

Lesson 1-3 Measuring Segments 23

Using the Midpoint

Algebra Qis the midpoint of PR.

What arePQ,QR, and PR?

Step 1 Findx.

PQ 5 Q R Definition of midpoint

6x2 7 5 5x1 1 Substitute.

x2 7 5 1 Subtract 5xfrom each side.

x5 8 Add 7 to each side.

Step 2 Find PQand QR.

PQ 5 6x2 7 QR 5 5x1 1

5 6(8) 2 7 Substitute 8 forx. 5 5(8) 1 1

5 41 Simplify. 5 41

Step 3 Find PR.

PR5 PQ 1 QR Segment Addition Postulate

5 41 1 41 Substitute.

5 82 Simplify.

PQand QRare both 41. PRis 82.

4. a. Reasoning Is it necessary to substitute 8 forxin the expression

for QR in order to nd QR? Explain.

b. Uis the midpoint of TV. What are TU, UV, and TV?

6x7

P Q R

5x1

8x11

T U V

12x1

How can you usealgebra to solve the

problem?The lengths of thecongruent segmentsare given as algebraicexpressions. You can setthe expressions equal toeach other.

Lesson Check

Do you know HOW?

Name each of the following.

1. Te point on DA)

that is 2 units from D

2. wo points that are 3 units from D

3. Te coordinate of the midpoint ofAG

4. A segment congruent to AC

Do you UNDERSTAND?

5. Vocabulary Name two

segment bisectors of PR.

6. Compare and Contrast Describe

the difference between saying that

two segments are congruentand saying that two

segments have equal length.When would you useeach phrase?

7. Error Analysis You and your friend live 5 mi apart.

He says that it is 5 mi from his house to your house

and 25 mi from your house to his house. What is

the error in his argument?

A B C D E F G

4 3 2 1 0 1 2 3 4

P Q R S T

2 3 4 5 6

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