GBK Geometry

1. GBK Geometry Jordan Johnson

2. Today’s plan • Greeting • Review Asg #17: From Ch. 3 Lesson 1 (pp. 80-83): • Set I Exercises 1-10, 18-22 • Set II Exercises 23-30, 36, 44-46 • Bonus: Set III • Lesson: Distance & Length • Homework / Questions • Clean-up

3. Ch. 3 Lesson 1, Set III If x = 1 and y = 1,then 2(x2 – y2) = 0 and 5(x – y) = 0,therefore 2(x2 – y2) = 5(x – y). By substitution, 2(x + y)(x – y) = 5(x – y). By division, 2(x + y) = 5. Substitution then gives 2(1 + 1) = 5and therefore, 2(2) = 5.

4. From the NY Times, 1999 How Is the Universe Built? Grain by Grain By GEORGE JOHNSON Published: Tuesday, December 7, 1999 Slightly smaller than what Americans quaintly insist on calling half an inch, a centimeter (one-hundredth of a meter) is easy enough to see. Divide this small length into 10 equal slices and you are looking, or probably squinting, at a millimeter (one-thousandth, or 10 to the minus 3 meters). By the time you divide one of these tiny units into a thousand minuscule micrometers, you have far exceeded the limits of the finest bifocals. But in the mind's eye, let the cutting continue, chopping the micrometer into a thousand nanometers and the nanometers into a thousand picometers, and those in steps of a thousandfold into femtometers, attometers, zeptometers, and yoctometers. At this point, 10 to the minus 24 meters, about one-billionth the radius of a proton, the roster of convenient Greek names runs out. But go ahead and keep dividing, again and again until you reach a length only a hundred-billionth as large as that tiny amount: 10 to the minus 35 meters, or a decimal point followed by 34 zeroes and then a one. You have finally hit rock bottom: a span called the Planck length, the shortest anything can get. According to recent developments in the quest to devise a so-called ''theory of everything,'' space is not an infinitely divisible continuum. It is not smooth but granular, and the Planck length gives the size of its smallest possible grains.

5. Length in the Real World

6. Length, Distance & Betweenness • Postulate 3 – The Ruler Postulate: • The points on a line can be numbered so that positive number differences measure distances. • Let a be A’s number, and let b be B’s number. • Then: AB = |a – b| • Example: if a = 40 and b = 54, AB = 14. B A

7. Length, Distance & Betweenness • Postulate 3 – The Ruler Postulate • Essentially, this says that any two points always have a measurable distance between them. • Useful when you have two points and need to discuss the distance between them. B A

8. Length, Distance & Betweenness • Definition: • A number associated with a point on a line is its coordinate.

9. Length, Distance & Betweenness • Definition: • A point is between two other points on the same line iff its coordinate is between their coordinates. • Abbreviation: • “A-B-C” means “B is between A and C on line AC.” • By definition, A-B-C iff a < b < c or a > b > c.

10. Length, Distance & Betweenness • A-C-B iff a < c < b or a > c > b. • According to the millimeter marks: • a = 40, b = 54, and c = 48.5. B C A

11. Betweenness of Points Theorem • Theorem: • If A-B-C, then AB + BC = AC. • Proof: (By definition of “between,” either a < b < c or a > b > c. Here we assume a < b < c; the proof is nearly identical for the case where a > b > c.) • A-B-C is given. • a < b < c by definition and our assumption. • AB = b – a and BC = c – b by the Ruler Postulate. • AB + BC = (b – a) + (c – b) = c – a by addition. • AC = c – a • AB + BC = AC by substitution.

12. Homework • Asg #18: Chapter 3, Lesson 2 (pp. 86-89): • Exercises #1-11, 16-28, 42-44, 47. • Bonus: Exercise 48 and Set III. • Due Monday, 10/15. • Asg #19: Chapter 3, Lesson 3 (pp. 93-97): • Exercises #1-7, 16-20, 34-43, 54 • Bonus: Set III. • Due Tuesday, 10/16.

13. Clean-up / Reminders • Pick up all trash / items. • Push in chairs (at front and back tables). • See you tomorrow!