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**LINES AND PLANES IN THREE DIMENSIONS**

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

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**ACTIVITY 1 : TO IDENTIFY THE PLANE**

G E F D C A B PLANE AT THE TOP : PLANE EFGH

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

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

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

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

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

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**THE LOCATION OF THE POINT**

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

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

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**ACTIVITY 2 : TO DETERMINE THE LOCATION OF A POINT**

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

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**ANGLE BETWEEN A LINE AND A PLANE**

C

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**Orthogonal projection**

Activity 3 :To Identify The Angle Between Line And Plane H G The line draw from G and perpendicular to the plane ABCD is call normal Normal E F D C 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. Orthogonal projection A B 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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**Angle between the line AG and the plane ABCD**

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

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 Diagram 2(a) shows a cuboid, ABCDEFG. Find the angle between the line AH and the plane DCGH.

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No Steps Solutions 1. Draw the line AH and shade the plan DCGH in diagram 2a. 12cm H G 5cm E F D C 4cm A B Diagram 2a Diagram 2a shows a cuboid, ABCDEFG. Find the angle between the line AH and the plane DCGH.

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No Steeps Solutions 2 Use the method you have learned in activity 3, identify the angle between the line AH and the plane DCGH A H D back 5cm B A C D E H G F 4cm 12cm

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No Steps Solutions 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 A D H 12cm G A H D 5cm E F D C 4cm A B

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No Steps Solutions 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 5 cm A D 4 cm H 12cm G 5cm E F D C 4cm A B

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**No Steps Solutions 6 H Mark, - the opposite side, AD asT**

- the adjacent side, HD as S 5 cm S A D 4 cm T 5cm B A C D E H G F 4cm 12cm

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**6 No Steps Solutions 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 Tan AHD = AHD = tan -1 AHD = ’ - SOH – CAH 5cm B A C D E H G F 4cm 12cm – TOA

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

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**ANGLE BETWEEN TWO PLANES**

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**DRAW 3 LINES ACTIVITY 5 : To Identified The Angle Between Two Planes**

EXAMPLE 3(a) G H E F D C DRAW 3 LINES A B Diagram 3a Diagram 3a shows a cuboid, ABCDEFGH. Name the angle between the plane AGH and the plane ABCD

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**ACTIVITY 5 : To Identified The Angle Between Two Planes**

Bottom A B Mark the location (direction) of the plane ABCD at the bottom of the first line to the left. Diagram 3a Diagram 3a shows a cuboid, ABCDEFGH. Name the angle between the plane, AGH and the plane, ABCD

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**ACTIVITY 5 : To Identified The Angle Between Two Planes**

Bottom A Refer to the plane, AGH, identify the points which touch the plane, ABCD and write it at the middle line. B Diagram 3a Diagram 3a shows a cuboid, ABCDEFGH. Name the angle between the plane, AGH and the plane, ABCD

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**ACTIVITY 5 : To Identified The Angle Between Two Planes**

H / G A E F D C Bottom 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. A B Diagram 3a Diagram 3a shows a cuboid, ABCDEFGH. Name the angle between the plane, AGH and the plane, ABCD

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**ACTIVITY 5 : To Identified The Angle Between Two Planes**

H/G A Bottom E F D C 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. A B Diagram 3a Diagram 3a shows a cuboid, ABCDEFGH. Name the angle between the plane, AGH and the plane, ABCD

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**ACTIVITY 5 : To Identified The Angle Between Two Planes**

Ke Bawah E F D C 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. A B Diagram 3a Diagram 3a shows a cuboid, ABCDEFGH. Name the angle between the plane, AGH and the plane, ABCD

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**ACTIVITY 5 : To Identified The Angle Between Two Planes**

Bottom E F D C Identify the point which is located at the bottom of the point H ( )and write it on the first line to the right. A B Diagram 3a Diagram 3a shows a cuboid, ABCDEFGH. Name the angle between the plane, AGH and the plane, ABCD

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**ACTIVITY 5 : To Identified The Angle Between Two Planes**

Bottom In the diagram 3a, complete the ∆ HAD and mark the HAD A B Diagram 3a Angle between the plane, AGH and the plane, ABCD = HAD

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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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