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OLEH PN. CHUANG BEE ENG SM TEKNIK BATU PAHAT

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LINES AND PLANES IN 3-DIMENSION To answer the question from this topic, the students must acquire the following skills : Able to identify the angle between a line and a plane ( 1 Mark ) Able to calculate the angle between a line and a plane ( 3 Marks ). Able to identify the angle between two planes ( 1 Mark ). Able to calculate the angle between two planes ( 3 Marks )

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Diagram shows a cuboid with a horizontal rectangular base. Calculate the angle between the planeTWR and the plane PSWT. R T / W S At the back W P V T U 8cm 5cm S R Q P1 EXAMPLE : Tan RWS = RWS = Tan -1 = 58 0 K2 N1 SOLUTION : W SR 8cm 5cm

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A B C D E F G H PLANE AT THE TOP :PLANE EFGH ACTIVITY 1 : TO IDENTIFY THE PLANE

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A B C D E F G H PLANE ON THE LEFT :PLANE ADHE

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A B C D E F G H PLANE IN THE FRONT :PLANE ABFE

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A B CD E F G H PLANE AT THE BACK :PLANE DCGH

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A B CD E F G H PLANE AT THE BOTTOM:PLANE ABCD

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A B CD E F G H PLANE ON THE RIGHT :PLANE BCGF

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ON TOP OF THE RED DOT TO THE RIGHT OF THE RED DOT IN FRONT OF THE RED DOT AT THE BACK OF THE RED DOT THE LOCATION OF THE POINT

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AT THE BOTTOM OF …. ON THE TOP OF …. AT THE BACK OF …. IN FRONT OF …. TO THE LEFT OF …. TO THE RIGHT OF ….

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ACTIVITY 2 : TO DETERMINE THE LOCATION OF A POINT A B C D E F G H POINT TO THE LEFT OF F : POINT E POINT G POINT AT THE BOTTOM OF F : POINT AT THE BACK OF F : POINT TO THE RIGHT OF D : POINT ON TOP OF D : POINT IN FRONT OF D : POINT B POINT C POINT H POINT A

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ANGLE BETWEEN A LINE AND A PLANE A B C LINE PLANE

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B A C D E F Orthogonal projection Normal H G Activity 3 :To Identify The Angle Between Line And Plane The line draw from G and perpendicular to the plane ABCD is call normal The line lies on the plane ABCD which joint the point A to the line GC is known as the orthogonal projection of line AG on the plane ABCD. The angle between the line AG and the orthogonal projection, AC is the angle between the line AG and the plane ABCD that is GAC.

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B A C D E F H G ACTIVITY 3 : To Identify The Angle Between A Line And A Plane A G C Angle between the line AG and the plane ABCD At the bottom Name the angle between the line AG and the plane ABCD Normal Orthogonal projection Example 1a = GAC.

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EXAMPLE 1(b) B A C D E H G F Diagram 1(b) Diagram 1b shows a cuboid ABCDEFGH. Name the angle between the line HB and the plane ABCD.

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Diagram 2(a) shows a cuboid, ABCDEFG. Find the angle between the line AH and the plane DCGH. ACTIVITY 4 : To find the angle between a line and a plane Example 2(a) 5cm B A C D E H G F 4cm 12cm Diagram 2a 5cm

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Draw the line AH and shade the plan DCGH in diagram 2a. 1. SolutionsStepsNo A C D E H G F 5cm 4cm 12cm B Diagram 2a Diagram 2a shows a cuboid, ABCDEFG. Find the angle between the line AH and the plane DCGH.

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NoSteepsSolutions 2 Use the method you have learned in activity 3, identify the angle between the line AH and the plane DCGH HA back D 5cm B A C D E H G F 4cm 12cm

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NoStepsSolutions 3 Refer to the points you have obtained in steep 2 (point A, H, D), complete the AHD. Mark AHD. Mark the right angle, HDA. Transfer out the AHD. H DA 5cm 4cm AH D B A C D E H G F 12cm

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NoStepsSolutions 4 With the information given in the question, label the length of the sides of AHD. At least the length for 2 sides must be known. Use Pythegoras Theorem if necessary. H D A 5 cm 4 cm 5cm B A C D E H G F 4cm 12cm

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NoStepsSolutions 6 Mark, - the opposite side, AD asT - the adjacent side, HD as S H DA 5 cm 4 cm S T 5cm B A C D E H G F 4cm 12cm

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NoStepsSolutions 6 Use the tangent formula to calculate AHD. Remember, use -The sine formula, if O and H were known - The cosine formula, if A and H were known -The tangent formula, if O and A were known Tan AHD = AHD = tan -1 AHD = – TOA - SOH – CAH 5cm B A C D E H G F 4cm 12cm

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example 2 (b) Diagram 2b shows a cuboid,ABCDEFGH. Calculate the angle between the line HB and the plane BCGF A B C D EF GH 4 cm 12 cm 3 cm Diagram 2b

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ANGLE BETWEEN TWO PLANES

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ACTIVITY 5 : To Identified The Angle Between Two Planes A E B C D H G F Diagram 3a Diagram 3a shows a cuboid, ABCDEFGH. Name the angle between the plane AGH and the plane ABCD 1.DRAW 3 LINES EXAMPLE 3(a)

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B A C D E H G F Diagram 3a Bottom 2.Mark the location (direction) of the plane ABCD at the bottom of the first line to the left. ACTIVITY 5 : To Identified The Angle Between Two Planes Diagram 3a shows a cuboid, ABCDEFGH. Name the angle between the plane, AGH and the plane, ABCD

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B A C D E H G F Diagram 3a A Bottom 3.Refer to the plane, AGH, identify the points which touch the plane, ABCD and write it at the middle line. ACTIVITY 5 : To Identified The Angle Between Two Planes Diagram 3a shows a cuboid, ABCDEFGH. Name the angle between the plane, AGH and the plane, ABCD

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B A C D E H G F Diagram 3a A H / G Bottom 3.Refer to the plane, AGH, identify the point which does not touch the plane, ABCD and write it at the first line to the left. ACTIVITY 5 : To Identified The Angle Between Two Planes Diagram 3a shows a cuboid, ABCDEFGH. Name the angle between the plane, AGH and the plane, ABCD

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B A C D E H G F Diagram 3a A H/G Bottom 5.Between the point H and G, point which is nearer to point A or located on the same plane as point A will be choosen. Point which is not choosen will be earased. ACTIVITY 5 : To Identified The Angle Between Two Planes Diagram 3a shows a cuboid, ABCDEFGH. Name the angle between the plane, AGH and the plane, ABCD

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B A C D E H G F Diagram 3a A H Ke Bawah 5.Between the point H and G, point which is nearer to point A or located on the same plane as point A will be choosen. Point which is not choosen will be earased. ACTIVITY 5 : To Identified The Angle Between Two Planes Diagram 3a shows a cuboid, ABCDEFGH. Name the angle between the plane, AGH and the plane, ABCD

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B A C D E H G F Diagram 3a A H Bottom D 6.Identify the point which is located at the bottom of the point H ( )and write it on the first line to the right. Diagram 3a shows a cuboid, ABCDEFGH. Name the angle between the plane, AGH and the plane, ABCD ACTIVITY 5 : To Identified The Angle Between Two Planes

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B A C D E H G F Diagram 3a Bottom A H D 7.In the diagram 3a, complete the HAD and mark the HAD Angle between the plane, AGH and the plane, ABCD = HAD ACTIVITY 5 : To Identified The Angle Between Two Planes

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EXAMPLE 3(b) Diagram 3b E 5cm B A C D H G F 4cm 12cm Diagram 3b shows a cuboid with horizontal rectangle base ABCD. Name the angle between the plane ACH and the plane CDHG

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THANK YOU

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