GEOMETRIC

1. GEOMETRIC POSITION OF A LINE TOWARD A PLANE

2. POSITION OF A LINE TOWARD A PLANE Kinds of possible position of a line toward other line in a plane: 1) h g Line g and line h is intersected 2) g h line g and line h is parallel

3. 3) g In a plane of a there is line g, then line h intersects plane aand line h doesn’t have point of intersection with line g. line g and lineh is crossed over POSITION OF A LINE TOWARDS OTHER LINES

4. Axioms of Two Parallel lines axiom 4 h A g Through a point outside the line, we only can make a line that parallel with the line. in the figure above, point A is outside line g. Through point A and line g, we can make a planea (look at Rule number 2, a plane is determined by a point and a line). Next, through point A, we can make lineh which parallel with lineg.

5. Rules of Two Parallel Lines Rule number 5 k Linek parallel with line l l Linel parallel with line m m Then line k is parallel with linem

6. Rule number 6 h linek parallel with lineh k linek intersect line g l line l parallel with line h g also intersect lineg Then, lines k, l, and g are in a plane Position of a Line Toward a Plane

7. Position of a Line Towards a Plane Line k parallel with line l k Line l intersect plane a Then line k intersect plane l

8. Position of a Line Towards a Plane 1) g A B Line g is in plane aif line g and plane a at least have 2 points of intersection (based on axiom 2,if a line and a plane have 2 points of intersection , then the whole line is in the plane)

9. Position of a Line Towards a Plane h Doesline h parallel with plane α? Line h is parallel with plane a, if line h and plane a doesn’t have any points of intersection.

10. Position of a Line Towards a Plane k Does Line k intersect plane α ?? Linekintersect plane α, if linekand plane αonly have a point of intersection.

11. Example: Let Cube ABCD EFGH g The edge of AB as the representative of lineg. The cube edges that intersect with line g is....... (AD, AE, BC, and BF) The cube edges that parallel with line g is.... (DC, EF,dan HG). The cube edges that cross over line g is..... (CG, DH, EH, and FG). Is there any cube edge that parallel with line g? (AB)

12. 2. Given cube The cube edges that in plane U is..... (AB, AD, BC, dan CD). The cube edges that parallel with plane U is..... (EF, EH, FG, andGH). The cube edges that intersect planeUis.... (EA, FB, GC, andHD).

13. Rules About lines Parallel with Plane Rules number 8 g h If line g parallel with line h and line h is on plane a, then line g parallel with plane a.

14. Rule number 9 g If plane a through line g and line g is parallel with plane β, then the intersection line of plane aand plane β will be parallel to line g POSITION OF A LINE TOWARD OTHER LINES

15. Rule number 10 g h a, If line g parallel with line h and line h parallel to plane a, then line g is parallel to plane a

16. POSITION OF A LINE TOWARDS OTHER LINES (a , β) Jika bidang a dan bidang β berpotongan dan masing-masing sejajar terhadap garis g, maka garis potong antara bidang a dan bidang βakan sejajar garis g. Rule number 11 If plane a and plane β intersected and each of them parallel to line g, then the intersection line between plane a and plane βwill be parallel with line g.

17. POSITION OF A LINE TOWARDS OTHER LINES Note: in rules number 9 and 11 need concept of intersection line between two planes. The concept of intersection line between two planes will be discussed in the next meeting.

18. Angle and Plane in Drawing Polyhedral • The Intersection of Line with Plane If there is a line and a point in a plane, then there will be 3 possibilities: 1. The line is in the plane if all points in the line is in that plane. 2. The line parallel with plane, if there is no point of intersection between line and plane. 3. The line intersected the plane, if it only has one point of intersection between line and plane. • Distance Between Points and Plane The distance of a point to a plane is the distance of this point to its plane projection.

19. Angle and Plane in Drawing Polyhedral • Angle Between Line and Plane The angle between line and plane is an angle between the line and its projection in a plane. D. Angle Between Two Planes Angles of two planes that intersected in line AB is an angle between two lines in a plane. Each of them are perpendicular to plane AB and intersect in one point.

20. Distances in Polyhedral • Given a cube ABCD.EFGH with edge length 8 cm. Points P,Q and R are in the mid points of edges AB,BC and plane ADHE respectively . Find the distance between: a. Points P and R b. Points Q and R c. Point H and line AC Answer : a. See that ∆PAR has a right angle on A AP = ½AB = 4 cm AR = ½AH =½ = PR = = = So, the distance points P and R is H G E F • R D C • S • Q • A B P

21. Angle Formed by line and a plane Example. Given a cube ABCD.EFGH with edge length 10 cm. a. Draw an angle between line AG and plane ABCD. b. Measure the angle size. Answer : a. Projection of line AG onto plane ABCD is line AC So, the angle between line AG and plane ABCD is GAC = b. See that CG = 10 cm and AC= 10 cm because AC is the diagonal of cube’s face. See that GAC has a right angle on C, then tan = or =35,30 G H E F D C A B Then, the angle size between line AG and plane ABCD is = 35,30

22. Example: Given cube ABCD.EFGH with edge length a single. Draw and find the angle between plane BDE and BDG Answer: Look at the following figure. The angle between plane BDE and plane BDG is a. See that ∆EPA is right angle in A,so that..

23. The End See You Next Meeting