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Three Dimension (Distance)

Three Dimension (Distance). After learning this slide, you’ll be able to determine the distance between the elements in the space of three dimension. We will study the distance : point to point point to line point to plane line to line line to plane, and plane to plane.

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Three Dimension (Distance)

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  1. Three Dimension (Distance) After learning this slide, you’ll be able to determine the distance between the elements in the space of three dimension

  2. We will study the distance : point to point point to line point to plane line to line line to plane, and plane to plane

  3. The distance of point to point This display, shows that the distance of point A to B is the length of line segment which connect point A to point B B Jarak dua titik A

  4. H G E F D C A B e. g. : Given that the edge length of a cube ABCD.EFGH is a cm. Determine the distance of : a) Point A to point C b) Point A to point G c) The distance of point A to the middle of plane EFGH P a cm a cm a cm

  5. H G E F a cm D C a cm A B a cm Solution: Consider Δ ABC which has right angle at B AC = = = = Thus, the diagonal of AC = cm

  6. H G E F a cm D C a cm A B a cm Distance of AG Consider Δ ACG which has right angle at C AG = = = = = Thus, the diagonal of AG = cm

  7. H G E F D C A B Distance of AP Consider Δ AEP which has right angle at E AP = = = = = Thus distance of A to P = cm P a cm

  8. Distance Point to Line A This display shows the distance from point A to line g is length of the line segment which is connected from point A and is perpendicular to line g. distance point to line g

  9. H G E F D C A B e.g. 1: Given that the edge length of a cube ABCD.EFGH is 5 cm. The distance from point A to the edge of HG is… 5 cm 5 cm

  10. H G E F D C A B Solutio The distance from point A to the edge of HG is length of the line segment AH, (AH  HG) 5 cm 5 cm AH = (AH is a side diagonal) AH = Thus, the distance from point A to the edge of HG= 5√2 cm

  11. H G E F D C A B e.g. 2: Given that the edge length of a cube ABCD.EFGH is 6 cm. The distance from point B to the diagonal of AG is… 6 cm 6 cm

  12. H G E F G 6√3 6√2 D P C A B B A 6 Solution The distance from point B to AG = the distance from point B to P (BP  AG) The side diagonal of BG = 6√2 cm The space diagonal of AG = 6√3 cm Consider a triangle ABG ! 6√3 cm P 6√2 cm 6 cm ?

  13. G 6√3 6√2 P B A 6 Consider a triangle ABG Sin A = = = BP = BP = 2√6 ? 2 Thus, the distance from point B to AG= 2√6 cm

  14. T D C A B e.g. 3 Given that T.ABCD is a pyramid. The edge length of its base is 12 cm, and the edge length of its upright is 12√2 cm. The distance from A to TC is... 12√2 cm 12 cm

  15. T 12√2 cm D C A 12 cm B Solution The distance from A to TC= AP AC is a cube’s diagonal AC = 12√2 AP = = = = Thus, the distance from A to TC= 6√6 cm 6√2 P 6√2 12√2

  16. H G E F D C A B e.g. 4 : Given that the edge length of a cube ABCD.EFGH is 6 cm and P 6 cm 6 cm Point P is in the middle of FG. The distance from point A to line DP is…

  17. P 3 cm G F H G E F D A 6 cm D C A B Solution  P Q 6√2 cm 6 cm R 6 cm DP = = =

  18. P 3 cm G F Q 6√2 cm D A R 6 cm Solution DP = Area of ADP ½DP.AQ = ½DA.PR 9.AQ = 6.6√2 AQ = 4√2 Thus the distance from point A to line DP= 4√2 cm 4

  19. V Perpendicular Line toward a plane Perpendicular line toward a plane means that line is perpendicular to two intersecting lines which are located on a plane.. g  a b g  a, g  b, Thus g  V

  20. V The Distance of a Point to a Plane This display shows the distance between point A and plane V is length of line segment which connect point A to plane V perpendicularly. A 

  21. H G E F D C A B e.g. 1 : Given that the edge length of a cube ABCD.EFGH is 10 cm. Thus the distance from point A to plane is…. P 10 cm

  22. H G E F D C A B Solution The distance from point A to plane BDHF is representated by the length of AP (APBD) AP = ½ AC (ACBD) = ½.10√2 = 5√2 P 10 cm Thus the distance from A to plane BDHF = 5√2 cm

  23. T D C A B e.g. 2 : Given that T.ABCD is a pyramid. The length of AB = 8 cm and TA = 12 cm. The distance from point T to plane ABCD is…. 12 cm 8 cm

  24. T D C A B Solution The distance from T to ABCD = The distance from T to the intersection of AC and BD= TP AC is a cube’ss diagonal AC = 8√2 AP = ½ AC = 4√2 12 cm P 8 cm

  25. T 12 cm P D C 8 cm A B AP = ½ AC = 4√2 TP = = = = = 4√7 Thus the distance from T to ABCD = 4√7 cm

  26. H G E F D C A B e.g. 3 : Given that the edge length of a cube ABCD.EFGH is 9 cm. The distance from point C to plane BDG is…. 9 cm

  27. H G E F D C A B Solution The distance from point C to plane BDG = CP That is the line segment which is drawn through point C and perpendicular to GT P T 9 cm CP = ⅓CE = ⅓.9√3 = 3√3 Thus the distance from C to BDG = 3√3 cm

  28. The Distance of line to line g This display explains the distance of line g and line hh is the length of line segment which connect those lines perpendicularly. P Q h

  29. H G E F D C A B e.g. Given that the edge length of a cube ABCD.EFGH is 4 cm. Determine the distance of: 4 cm • Line AB to line HG • Line AD to line HF • Line BD to line EG

  30. H G E F D C A B Solution The distance of line: AB to line HG = AH (AH  AB, AH  HG) = 4√2 (a side diagonal) b.AD to line HF = DH (DH  AD, DH  HF = 4 cm 4 cm

  31. H G E F D C A B Solution The distance of: b.BD to line EG = PQ (PQ  BD, PQ  EG = AE = 4 cm Q P 4 cm

  32. g V The Distance of Line to Plane g This display shows the distance of line g to plane V is length of line segment which connect that line and plane perpendicularly.

  33. H G E F D C A B e.g. 1 Given that the edge length os a cobe ABCD.EFGH is 8 cm The distance of line AE to plane BDHF is…. P 8 cm

  34. H G E F D C A B Solution The distance of line AE to plane BDHF Is represented by the length of AP.(AP AE AP  BDHF) AP = ½ AC(ACBDHF) = ½.8√2 = 4√2 P 8 cm Thus the distance from A to BDHF = 4√2 cm

  35. V W W • The Distance of Plane to Plane This display explains the distance of plane W and plane V is length of line segment which is perpendicuar to plane W and is perpendicular to plane V. Jarak Dua Bidang

  36. H G E F D C A B e.g. 1 : Given that the edge length of a cube ABCD.EFGH is 6 cm. The distance of plane AFH to plane BDG is…. 6 cm 6 cm

  37. H G E F D C A B Solution The distance of plane AFHto plane BDG Is represented by PQ PQ = ⅓ CE (CE is a space diagonal) PQ = ⅓. 6√3 = 2√3 Q 6 cm P 6 cm Thus the distance of AFH to BDG = 2√3 cm

  38. H G E F D C A B e.g. 2 : Given that the edge length of a cube ABCD.EFGH is 12 cm. M L K 12 cm Points K, L and M are the middle point of BC, CD dan CG. The distance of plane AFH and KLM is….

  39. H G E F D C A B Solution • Diagonal EC = 12√3 • The distance from E to AFH = distance from AFH to BDG = distance from BDG to C L 12 cm Thus the distance from point E to AFH = ⅓EC =⅓.12√3 = 4√3 So that the distance from BDG to C is 4√3 too.

  40. H G E F D C A B The distance of BDG to point C is 4√3. The distance of BDG to KLM = distance of KLM to point C = ½.4√3 = 2√3 M L K 12 cm Thus the distance of AFH to KLM = Distance of AFH to BDG + distance of BDG to KLM = 4√3 + 2√3 = 6√3 cm

  41. Have a nice try !

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