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Understanding Points, Lines, and Planes in Geometry

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daquan-west
17 diapositives · 293 Views · Nov 17, 2014

This lesson covers the fundamental concepts of points, lines, and planes in geometry. A point is defined as an entity without size, represented as a dot, and labeled with capital letters. Lines extend infinitely without thickness, can be named in various ways, and the relationship between lines and points is explored through postulates. The lesson discusses collinear points, planes as flat surfaces, and coplanar objects. We also examine the intersections of lines and planes, providing essential postulates and definitions for better understanding geometric relationships.

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  1. Lesson 1-1 Point, Line, Plane Lesson 1-1 Point, Line, Plane

  2. Definition • A postulate or axiom is an accepted statement of fact. Lesson 1-1 Point, Line, Plane

  3. Points • Points do not have actual size. • How to Sketch: Using dots • How to label: Use capital letters Never name two points with the same letter (in the same sketch). A B A C Lesson 1-1 Point, Line, Plane

  4. Lines • Lines extend indefinitely and have no thickness or width. • How to sketch : using arrows at both ends. • How to name: 2 ways (1) small script letter – line n (2) any two points on the line - • Never name a line using three points - n A B C Lesson 1-1 Point, Line, Plane

  5. Postulate 1-1 • Through any two points there is exactly one line. Lesson 1-1 Point, Line, Plane

  6. Intersection of Figures The intersection of two figures is the set of points that are common in both figures. The intersection of two lines is a point. m Line m and line n intersect at point P. P n Continued……. Lesson 1-1 Point, Line, Plane

  7. Postulate 1-2 • If two distinct lines intersect, then they intersect at exactly one point. Lesson 1-1 Point, Line, Plane

  8. Collinear Points • Collinear points are points that lie on the same line. (The line does not have to be visible.) • A point lies on the line if the coordinates of the point satisfy the equation of the line. Ex: To find if A (1, 0) is collinear with the points on the line y = -3x + 3. Substitute x = 1 and y = 0 in the equation. 0 = -3 (1) + 3 0 = -3 + 3 0 = 0 The point A satisfies the equation, therefore the point is collinear with the points on the line. A B C Collinear C A B Non collinear Lesson 1-1 Point, Line, Plane

  9. Planes • A plane is a flat surface that extends indefinitely in all directions. • How to sketch: Use a parallelogram (four sided figure) • How to name: 2 ways (1) Capital script letter – Plane M (2) Any 3 non collinear points in the plane - Plane: ABC/ ACB / BAC / BCA / CAB / CBA A M B C Horizontal Plane Vertical Plane Other Lesson 1-1 Point, Line, Plane

  10. Postulate 1-4 • Through any three noncollinear points there is exactly one plane. Lesson 1-1 Point, Line, Plane

  11. Different planes in a figure: A B Plane ABCD Plane EFGH Plane BCGF Plane ADHE Plane ABFE Plane CDHG Etc. D C E F H G Lesson 1-1 Point, Line, Plane

  12. Other planes in the same figure: Any three non collinear points determine a plane! Plane AFGD Plane ACGE Plane ACH Plane AGF Plane BDG Etc. Lesson 1-1 Point, Line, Plane

  13. Coplanar Objects Coplanar objects (points, lines, etc.) are objects that lie on the same plane. The plane does not have to be visible. Are the following points coplanar? A, B, C ? Yes A, B, C, F ? No H, G, F, E ? Yes E, H, C, B ? Yes A, G, F ? Yes C, B, F, H ? No Lesson 1-1 Point, Line, Plane

  14. Intersection of Figures The intersection of two figures is the set of points that are common in both figures. The intersection of two lines is a point. m Line m and line n intersect at point P. P n Continued……. Lesson 1-1 Point, Line, Plane

  15. 3 Possibilities of Intersection of a Line and a Plane (1) Line passes through plane – intersection is a point. (2) Line lies on the plane - intersection is a line. (3) Line is parallel to the plane - no common points. Lesson 1-1 Point, Line, Plane

  16. Intersection of Two Planes is a Line. B P A R Plane P and Plane R intersect at the line Lesson 1-1 Point, Line, Plane

  17. Postulate 1-3 • If two distinct planes intersect, then they intersect in exactly one line. Lesson 1-1 Point, Line, Plane