9
Chapter I review
Textbook: pp. 22–23
GoalReview the notions, axioms and properties of Chapter I.
New words
axiom · aksiomahalf-plane · yarimtekislikmidpoint · kesma o‘rtasilocus · geometrik o‘rin
Explanation
Main notions of the chapter: point, line, plane (undefined), segment, ray, half-plane, circle, disc. Axioms: one line passes through two points; of three points on a line one lies between; a line divides the plane into two half-planes; on a ray one segment equal to a given segment can be laid off; if B is between A and C then AC = AB + BC. Remember: n points (no three on a line) give n(n − 1) : 2 lines. A diameter is d = 2r. When solving, first draw a figure and check whether the order of points can vary.
Worked examples
5 points, no three on a line: 5 · 4 : 2 = 10 lines.
If AB = 5 and BC = 3 (A, B, C on one line), then AC = 8 or AC = 2.
Class activity
“Chapter quiz”: two teams take turns stating a definition or axiom; the other team draws a matching figure.
Practice
1
A, B, C, D lie on a line in this order. AB = 3, BC = 4, CD = 5. AD = ?
12
2
7 points, no three collinear: how many lines are there?
21
3
On a 15 cm segment AB, point C is such that AC = 2 · CB. Find AC (cm).
10
4
Why do we not prove “one line passes through two points”?
It is an axiom: a basic property accepted without proof.