2
The simplest figures: point, line and plane
Textbook: pp. 8–9
GoalKnow point, line and plane and their first axioms.
New words
point · nuqtaline · to‘g‘ri chiziqplane · tekislikhalf-plane · yarimtekislik
Explanation
Point, line and plane are the basic notions of geometry and are not defined. Points are named with capital Latin letters (A, B, C) and lines with small letters (a, b, c); “point A belongs to line c” is written A ∈ c. Axiom: for any line in the plane there are points that lie on it and points that do not. Axiom: exactly one line passes through any two points. Axiom: every line divides the plane into two half-planes, and the line itself is their common boundary. If two lines have a common point, they intersect at that point.
Worked examples
Two points A and B are given. Exactly one line AB passes through them, so a line can also be named by two of its points.
If three points do not lie on one line, three lines pass through pairs of them: AB, BC and AC.
Class activity
“String and pegs”: two pupils hold a string tight — it shows the one line through two points. A third pupil stands off the string, showing a point that does not belong to the line.
Practice
1
Four points are given, no three on one line. How many lines pass through pairs of them?
6
2
How many common points can two different lines have?
0 or 1
3
Three lines in a plane intersect in pairs and do not all pass through one point. How many intersection points are there?
3
4
Why are two points enough to determine a line?
Because by the axiom exactly one line passes through two points.