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1.3 Segments and Their MeasuresPowerPoint Presentation

1.3 Segments and Their Measures

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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.

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.

Names of points

A

B

x1

x2

Coordinates of points

Postulate 1: Ruler PostulateA

AB

B

x1

x2

AB = |x2 – x1|

Measure the length of the segment to the nearest millimeter. numbers. The real number that corresponds to a point is the

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.

Ex. 1: Finding the Distance Between Two PointsAB = |5.5 – 3| = |2.5| = 2.5

Note: numbers. The real number that corresponds to a point is the

- 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 numbers. The real number that corresponds to a point is the

Finding the Distance on a Number Line

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

Solutions:

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

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.

Note:A

Point B is between points A and C.

B

C

D

E

Point E is NOT between points D and F.

F

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

Postulate 2: Segment Addition PostulateAC

A

B

C

AB

BC

USING SEGMENT POSTULATES AC, then B is between A and C.

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

The Distance Formula AC, then B is between A and C.

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

B(x2, y2)

A(x1, y1)

USING THE DISTANCE FORMULA AC, then B is between A and C.

What is the Distance Formula based on?

B(x2, y2)

| y2 – y1 |

| x2 – x1 |

C(x2, y1)

A(x1, y1)

Pythagorean Theorem

c2 = a2 + b2

D =

USING THE DISTANCE FORMULA AC, then B is between A and C.

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

Solution:

Classwork AC, then B is between A and C. p. 21-22 #19, 31, 35, 37, 41

p. 21-22 #14 -42, 46, 60-70 Evens AC, then B is between A and C.

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