SMK (P) Bukit Kuda

1. SMK (P) Bukit Kuda Liyana Athirah Hooi Cheng

2. MATHEMATICS LINES AND PLANES IN 3 DIMENSIONS FORM 4

3. LINES AND PLANES IN 3-DIMENSIONS Introduction Prior knowledge Line Plane The angle between a line and a plane The angle between two planes Conclusion Exit

4. Prior Knowledge A.Different types of dimensions One- Dimensional Two- Dimensional Three- Dimensional B.Pythagoras Theorem Sine = Opposite Hypotenuse Cosine = Adjacent Hypotenuse Tangent = Opposite Adjacent

5. PLANES Definition: A completely flat surface HORIZONTAL PLANE VERTICAL PLANE INCLINED PLANE

6. VERTICAL PLANE HORIZONTAL PLANE INCLINED PLANE

7. D C Q P A B LINE LIES ON A PLANE

8. X D C A B Y LINE INTERSECTS WITH A PLANE

9. The angle between a line and a plane Objective: To find and calculate the angle between a line a plane • Steps to find the angle: • identify a normal to a plane • determine the orthogonal projection of the line • name the angle • calculate the angle

10. X S R Y P Q NORMAL TO A PLANE Definition: Normal to a plane is a straight line which is perpendicular to the intersection of any lines on the plane. Normal to a plane

11. B S R A C P Q Orthogonal projection Based on the following diagram The Orthogonal projection is a line which joins point A with point C. The line lies on the plane PQRS. BC is normal to the plane PQRS AC is the orthogonal projection of line AB on a plane PQRS. The angle between the line AB and the plane PQRS is < BAC

12. S R Q P 4 cm D C B 3 cm A The angle between a line and a plane Example Based on the above diagram, find the angle between the line PB and the plane ABCD.

13. S R P Q 4 cm D C 3 cm A B The angle between a line and a plane Name the angle The angle between the line PB and the plane ABCD is < ABP How to calculate the angle ?

14. S R P Q P 3 cm D C 3cm A B 4 cm A B 4 cm How to calculate the angle ? Find the angle of ABP, if AB = 4 cm, PA = 3 cm Tan < ABP = AP AB = 3 4 = 0.75 Use a scientific calculator to find the answer < ABP = tan –1 0.75 = 36°52 ́

15. S R P Q D C A B S 19 cm D B 13 cm The angle between a line and a plane Find the angle between the line SB and the plane ABCD. Answer : The angle is < DBS Calculate the angle of < DBS if SB = 19cm and BD= 13 cm. Cos < DBS = DB SB = 13 19 = 0.6842 < DBS = cos –1 0.6842 = 46°49 ́

16. Exercise Answers S R P Q D C A B Based on the diagram,name the angles between the following: • Line BR and plane ABCD • Line AS and plane ABCD • Line AR and plane CDSR • Line BS and plane PQRS (a) ∠RBC (b) ∠SAD (c) ∠ARC (d) ∠BSQ

17. E D C F 5 cm 9 cm A B 12 cm AF and DE are both perpendicular to the plane ABCD.AF=DE=5cm, AB=DC= 12cm,AD=BC=9cm,and ABCD is a rectangle. Calculate the angle between (i) the line BF and the plane ABCD; (ii)the line BF and the plane ADEF ; (iii) the line FC and the plane ABCD. Answers

18. Answers • Since AF is perpendicular to the plane ABCD , the angle between the line • BF and the plane ABCD is given by the angle ABF. ⇒tan ∠ABF = 5 12 = 0.425 ⇒ ∠ABF = 23º Thus the angle between the line BF and the plane ABCD is 23º (ii) AF is perpendicular to plane ABCD ⇒ AF is perpendicular to AB. ⇒ The angle between the line BF and the plane ADEF is given by angle ∠AFB ∠AFB = 90º - ∠ABF = 90º - 23º = 67º Thus the angle between the line BF and the plane ADEF is 67º No(iii)

19. (iii) Since AF perpendicular to the plane ABCD,the angle between FC and ABCD is given by the angle ACF.By pythagoras ’ theorem, AC² = 12² + 9² ⇒ AC = 15cm in ∆ACF,tan ∠ACF = AF AC = 5 13 ⇒ ∠ACF = 21°.

20. A B D C E F The Angle Between Two Planes Intersection Line The angle between two planes which meet on a line DC is the angle between two lines, one in each plane,taken perpendicular to DC and meeting at a point on DC .

21. Example 1 S R P Q D C Steps to find the angle • Identify the intersection line between • two planes ( ie SD) • Identify perpendicular lines from plane • ADSP and plane BDS to the intersection line • (ie PS/AD and BD respectively) • Identify the perpendicular line which meet • at the same point on the intersection line • (AD and BD meet at the point) • Name the angle • (the angle is ∠ ADB) A B Name the angle between planes ADSP and BDS

22. Example 2 6 cm P Q 16 cm B A R S N 6 cm D C • The diagram below shows a cuboid with a rectangular base ABCD. • N is the mid-point of AD. Calculate • Find the length of NC • Calculate the angle between RN and the base • Name the angle between plane RQAN and plane PQRS Answers

23. N 8 cm D C 6 cm R 6 cm C N 10 cm Back to question (a) AD = QR = 16 cm ND = ½AD = 8 cm DC = PQ = 6 cm NC²= 8² + 6² = 100 NC = √100 = 10 cm (b) The angle between RN and the base ABCD is ∠RNC. tan ∠RNC = 6/10 = 0.6 ∠RNC = 30°58’ (c) The angle between planes RQAN and PQRS is ∠AQP.

24. Exercise Answers Z Y W X G H E F • EFGH,WXYZ is a cube.Name the angle between; • plane EFYZ and base EFGH; • plane EFYZ and plane WXYZ; • plane EFGH and plane HGYZ ; • plane FHZX and plane HGYZ. • ∠YFG, ∠ZEH (ii) ∠XYF, ∠WZE (iii)∠EHZ, ∠FGY (iv) ∠YZH, ∠FHG

25. P Q E F R S H G Multiple Choice Question: Based on the above diagram, which of the following line is not intersect with plane PQRS? A FG B FQ C HS D PG

26. Q P R S T U Based on the above diagram,which of the following line is normal to the plane? A PF B PQ C PS D ST

27. P Q R S E F H G Based on the above diagram, orthogonal projection for line PG to the Plane SRGH is A PS B RS C SG D SH

28. P Q R S E F H G Based on the above diagram, the angle between line SF and the plane SRGH is A ∠ FSG B ∠ FSH C ∠ SEH D ∠ SFG

29. P Q R S E F H G The angle between PR and plane PESH is A 5º43’ B 35º16’ C 45º D 54º44’

30. Q P R S E F H G Based on the above diagram, the angle between plane PQHG and EFGH is A ∠ PGE B ∠ QGF C ∠ QHF D ∠ SPH

31. LINES AND PLANES IN 3-DIMENSIONS • LINES • lies on a plane • intersect with a plane PLANES - Horizontal - inclined - Vertical • ANGLE BETWEEN A LINE ANDA PLANE • Normal to plane • Orthogonal projection • ANGLE BETWEEN TWO PLANES • Lines of intersection • Perpendicular line to the line ofintersection • CALCULATE THE ANGLE • determine right triangle • use sine, cosine or tangent to calculate the angle