1.3 Segments and Their Measures

Objectives/Assignment

• Use segment postulates.• Use the Distance Formula to measure

distances as applied.

Using Segment Postulates

• In geometry, rules that are accepted without proof are called postulates, or axioms. Rules that are proved are called theorems. In this lesson, you will study two postulates about the lengths of segments.

Postulate 1: Ruler Postulate

• The points on a line can be matched one to one with the real numbers. The real number that corresponds to a point is the coordinate of the point.

• The distance between points A and B written as AB, is the absolute value of the difference between the coordinates of A and B.

• AB is also called the length of AB.

A B

x1 x2

Names of points

Coordinates of points

A B

x1 x2

AB

AB = |x2 – x1|

Ex. 1: Finding the Distance Between Two Points• Measure the length of the segment to the nearest millimeter.Solution: Use a metric ruler. Align one mark of the ruler with A.

Then estimate the coordinate of B. For example if you align A with 3, B appears to align with 5.5.

The distance between A and B is about 2.5 cm.

AB = |5.5 – 3| = |2.5| = 2.5

Note:

• It doesn’t matter how you place the ruler. For example, if the ruler in Example 1 is placed so that A is aligned with 4, then B align 6.5. The difference in the coordinates is the same.

USING SEGMENT POSTULATES

Finding the Distance on a Number Line

Example 1

Solutions:AB = 1.5, BA = 1.5, CB = 3, BC = 3, BD = 5, DB = 5

Find AB, BA, CB, BC, BD, and DB

Note:

• When three points lie on a line, you can say that one of them is between the other two. This concept applies to collinear points only. For instance, in the figures on the next slide, point B is between points A and C, but point E is not between points D and F.

A

B

C

Point B is between points A and C.

D

F

E

Point E is NOT between points D and F.

Postulate 2: Segment Addition Postulate

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

AC

A B CAB BC

USING SEGMENT POSTULATES

Example 2

A DB CE

In the diagram, AD = 34, BC = 14, CD = 13 and AB = BE = EC.

Find BE, EC, AB, AE, ED, and BD

Solution: BE = 7, EC = 7, AB = 7, AE = 14, ED = 20, and BD = 27

2)12(2)12( yyxxd

The distance d between two points (x1,y1) and (x2,y2) is given by

The Distance Formula

A(x1, y1)

B(x2, y2)

USING THE DISTANCE FORMULA

What is the Distance Formula based on?

cb

aA(x1, y1) C(x2, y1)

B(x2, y2)

| y2 – y1 |

| x2 – x1 |

Pythagorean Theorem

c2 = a2 + b22

122

12 )()( yyxx D =

USING THE DISTANCE FORMULA

Example 3

Find the distance between the points X(2, -5) and Y(- 4, - 9)

Solution: 2111.713252

Classwork p. 21-22 #19, 31, 35, 37, 41

p. 21-22 #14 -42, 46, 60-70 Evens