Lesson 3.2 Segment Measure pp. 89-92

Objectives: 1. To identify and apply the Ruler Postulate. 2. To define distance and use coordinates to determine distances. 3. To define between using distance. 4. To explain the Completeness Postulate and its connection to the Ruler Postulate.

Postulate 3.1 Ruler Postulate. Every point of a line can be placed in correspondence with a real number. -1 -2 -3 1 2 3

Definition The coordinate of a point on a line is the number that corresponds to the point. -1 -2 -3 1 2 3 A B C

Definition The distance between two points A and B is the absolute value of the difference of their coordinates. Distance between points A and B is denoted by AB, given by AB = |a - b|.

EXAMPLES Find the distances. 1. BD 2. AE 3. CE 5 -5 A B C D E Answers 1. BD = |-5 – 2| = |-7| = 7 2. AE = |-8 – 3| = |-11| = 11 3. CE = |-1 – 3| = |-4| = 4

Definition B is between A and C if BC  BA = {B} when A, B, and C are collinear. In symbols, you can write A-B-C.

Definition A point M is Between A and B if AM + MB = AB. The correct notation is A-M-B.

Postulate 3.2 Completeness Postulate. Given a ray, AB, and any positive real number r, there is exactly one point C on the ray so that AC = r.

Practice: Graph 12 on a number line. -1 3 4 5

Practice: What is the distance between 64 and -13?

Practice: Does it matter which one is considered the first point in the previous question?

Homework pp. 91-92

►A. Exercises 11. CK CK = |0 – 9| = |-9| = 9 X A F C L K T -13 -10 -5 0 5 10 12 X A F C L K T CK = |0 – 9| = |-9| = 9

►A. Exercises 13. TX TX = |12 – (-13)| = |25| = 25 X A F C L K T -13 -10 -5 0 5 10 12 X A F C L K T TX = |12 – (-13)| = |25| = 25

17. Show that C is between A and T. ►B. Exercises 17. Show that C is between A and T. -13 -10 -5 0 5 10 12 X A F C L K T According to the definition, C is between A and T if AC + CT = AT. AC = 8, CT = 12, and AT = 20. Since 8 + 12 = 20, therefore A-C-T.

►B. Exercises Find the distance between two points with the given coordinates. 21. 47 and -12 |47 – (-12)| = |47 + 12| = |59| = 59

■ Cumulative Review 28. If B is between A and C, which is longer: AB or AC?

■ Cumulative Review 29. Can a half-line contain a ray?

■ Cumulative Review 30. For sets A and B, which statements are always true? a. A  A b. If A  B, then B  A c. If A  B and B  C, then A  C

■ Cumulative Review 31. Is the subset relation reflexive? transitive?

■ Cumulative Review 32. Is the subset relation an equivalence relation?

Analytic Geometry The Distance Formula

Distance Formula. The distance, d, between two points A (x1, y1) and B (x2, y2) is d = (x1 - x2)2 + (y1 - y2)2.

Find the distance between (-1, -2) and (-3, 4). d = (x1 – x2)2 + (y1 – y2)2 = (-1 – [-3])2 + (-2 – 4)2 = 22 + (-6)2 = 2 10 ≈ 6.32

Find the distance between (6, 2) and (0, 3).

Find the distance between (3, 5) and (7, 5).

Use the distance formula to determine betweenness. A (1,11), B (3, 8), C (7, 2)