Lines and Planes in 3d

Lines and Planes in

20

3-Dimensions

4.

Paper 1 J

1.

M

M

L K

J H

K

G

L H

G

E

E

F

F

The angle between line JG and the base is ∠ JGE . Answer: C

is an inclined plane. FGJK is Answer: C

5. M L

J

2.

K

T

H

S

R

E

G

O

F

P

Q

The angle between line LE and plane EHMJ is is ∠ MEL.

The orthogonal projection of line TQ onto plane PQRS is OQ.

Answer: D

Answer: B

6. T

3.

T

S R

Q

R N

P

M

Q

The angle between line TQ and the base PQRS is ∠TQN .

P

The angle between line TM and the plane PQR is ∠TMR.

Answer: A

Answer: A

1

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Penerbitan Pelangi Sdn. Bhd.

Mathematics

SPM

Chapter 20

7.

2.

H

E

G

D

9 cm

M

x

C

c m 1 0

F E

A

The angle between plane GEH and plane GEF is ∠ HMF .

B

9 tan x = ––– 10 x = 41°59′

Answer: A

The angle between line EC and and plane ABCD is 41°59′. 8.

J

M

3. K

J

L

L

K

H

E

M

G

10 cm

F

H

G

x

5 cm

The angle between plane FGJK and and plane EFGH is ∠ EFK or or ∠ HGJ .

E

12 cm

F

HF 2 = HE 2 + EF 2

Answer: C

52 + 122 = 13 cm

HF =

9. W

V

JH HF

tan x = –––– T

10 = ––– 13 x = 37°34′

U S

R

P

The angle between line JF and the base EFGH is 37°34′.

Q

The angle between plane PQV and the base PQRS is ∠ RQV .

4.

T

Answer: D

Paper 2 24 cm

1.

J

8 cm

x

H 12 cm

P

G

x 9 cm

E

S

F

QS 2 = SR2 + RQ2

8 tan x = ––– 12 x = 33.7°

QS =

=

The angle between line JG and plane EFGH is is 33.7°.

©


2

72 + 7 2 98 cm

7 cm R

Q 7 cm

Mathematics

SPM

Chapter 20

8.

TQ

tan x = ––––

T

QS

24 = –––– 98

5 cm

x = 67°35′

P

R x

The angle between line ST and the base PQRS is 67°35′.

M

10 cm

6 cm

Q

5.

PM 2 = PQ2 – QM 2

H

m c 7

PM = 102 – 62 12 cm

= 8 cm

G

TP PM

tan x = –––– E

5 =— 8 x = 32°

7 12

sin ∠ HFG = ∠ HFG =

F

The angle between plane TQR and the base PQR is 32°.

�

35°41

The angle between line HF and the base EFG is 35°41 . �

6.

E

H

Paper 1 F

G

1.

D

C

6 cm

8 cm

H

B

K

6 tan x = — 8 x = 36°52′

G

F

The angle between plane FHJ and plane KHJ is ∠KMF .

E

Answer: D

F

A

N

E

The angle between plane BCEF and and the base ABCD is 36°52′.

7.

M

7 cm

x A

J

4 cm

D 7 cm

x

C

B 3 cm

4 tan x = — 3 x = 53°8′ The angle between plane ABFE and and the base ABCD is 53°8′.

3

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Mathematics

SPM

Chapter 20

2.

6. J

J

M L

K

U

E

F

V

H

T

G

E

L

K

H

M

G

F

The angle between plane KGJ and the base EFGH is ∠ JGH .

The angle between the plane FTL and the plane FGML is ∠TUV .

Answer: B

Answer: D

3.

7.

Q

L

P

T R P

R

Q

M

N

N

L M

S

The angle between line TN and the base LMNP is ∠TNL.

T

The angle between plane PRSN and plane LMNP is ∠SNM .

Answer: A

Answer: A

4. R

U

Paper 2

T

S Q

M

1. P E

H

M

N

G F

The angle between the plane SNP and the plane PNTU is ∠SNT .

7 cm

C 3 cm

N

Answer: D A

5.

10 cm

MN NL

(b) tan ∠ MLN = ––––

S

7 = –– 10

Q P

∠ MLN

The angle between line TS and the base PQRS is ∠TSQ. Answer: B


L 3 cm B

(a) The an angle gle between between the plane MBC and and the base ABCD is ∠ MLN .

T

R

©

D

4

= 35°

Mathematics

2.

S

SPM

Chapter 20

5.

R

F

E

M 5 cm

7 cm

Q

P

x

A

D

8 cm 15 cm

Y

16 cm

B

Z

C

10 cm 12 cm

W

ED AD

tan x = ––––

X

7 = ––– 16 x = 23°38′

(a) The angle between line XM and and plane WZSP is ∠ XMW . (b)

The angle between plane EAB and the base ABCD is 23°38′.

MW 2 = PM 2 + PW 2 MW =

52 + 152

= 250 cm

6.

(a)

E

WX tan ∠ XMW = –––– MW

12 = ––––– 250 ∠ XMW

F D

7 cm

A

3.

C

6 cm

= 37.2°

B

8 cm

E 6 cm H

G

F

D

7 cm

The angle between the plane BCEF and and the base ABCD is ∠ ABF or ∠ DCE .

C 15 cm

A

10 cm

(b)

6 8

tan ∠ ABF =

B

∠ ABF

(a) The angle between the the line CE and the plane ADEF is is ∠CED.

= 36°52′

7. J

(b)

9 cm

10 = 7 ∠CED

H

= 55°

(a)

E

H

D

152 + 8 2 = 17 cm

C

KG EG

tan x = –––– 10 cm

A

9 = ––– 17 x = 27°54′

B

The angle between the plane DFG and the plane is ∠ AFD. ABGF is (b) tan ∠ AFD = ∠ AFD

F

15 cm

EG =

G

4 cm

8 cm

EG2 = EF 2 + FG2

7 cm

F

G

x E

4.

K

CD tan ∠CED = ED

The angle between line EK and the base EFGH is ∠KEG and ∠KEG = 27°54′.

10 4

= 68°12′

5

©


Mathematics

SPM

Chapter 20

5.

X

W

Paper 1

T

1.

V

U S

K

R

J

P

H E

Q

The angle between plane PWV and plane RSWV is ∠PWS .

G F

Answer: A

The angle between line FK and and plane GHK is is ∠GKF . Answer: D

6.

2.

M

V

L

J

R

K N

P X H

Q

G

E

The angle between the base PQR and plane VPQ is ∠ RXV .

F

Answer: B

The angle between line FN and and plane EHMJ is is ∠ ENF . Answer: A

7.

3.

T

U

H

S

R

P E

Q

G

The angle between plane QST and plane RSTU is ∠ RSQ.

M

Answer: B

F

The angle between line HG and plane EFH is is ∠GHM .

8.

V

Answer: C H

T

4.

G J

M

S

P

R

E

The angle between plane VFG and the base EFGH is ∠VJM .

Q

The angle between line QT and and plane PST is is

Answer: C

∠PTQ.

Answer: D

©


F

6

Mathematics

9.

M X

1.

J

Chapter 20

Paper 2

L

K

SPM

V

G

H Y

E

F 12 cm

The angle between plane XFGM and and plane JKLM is ∠GML.

3 cm

Answer: D

C

D

10.

T

6 cm

A

U

B

BM 2 = BC 2 + CM 2

R

S

x

M

4 cm

BM =

42 + 32 = 5 cm

X

Y P

Q

VM BM

tan x = –––– The angle between plane UPS and the base PQRS is ∠UYX .

12 = ––– 5 x = 67°23′

Answer: A 11.

M

The angle between line VB and the base ABCD is ∠VBM and and ∠VBM = = 67°23′.

L K

J

R

2.

Q

V 13 cm

P G

H

D

E

C

F

The angle between plane PGL and plane HGLM is ∠ RQP.

M x A

12 cm

12 cm

B

Answer: B BD2 = BC 2 + CD2 12.

N

BD = 122 + 122

= 288 cm M

P

Q J

288 2 = 8.485 cm

BM = –––––

L

VM BM

tan x = ––––

K

13 = ––––– 8.485 x = 56°52′

The angle between plane NKL and plane NJM is ∠PNQ. Answer: C

The angle between line VB and the base ABCD is ∠VBM and and ∠VBM = = 56°52′.

7

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Mathematics

SPM

Chapter 20

3.

5.

V

J

M

8 cm

L

K D

C 6 cm

x A

12 cm

9 cm

B

m c 5 H

E

G x F

12 cm

BD2 = BC 2 + CD2 FH 2 = HE 2 + EF 2

BD =

92 + 122 = 15 cm

FH =

52 + 122 = 13 cm

VD BD

tan x = ––––

JH FH

8 = ––– 15 x = 28°4′

tan x = –––– 6 = ––– 13

The angle between line VB and the base ABCD is ∠VBD and ∠VBD = 28°4′.

x = 24°47

�

The angle between line JF and the base EFGH is 24°47 . �

4. V

6.

H

4 cm E

9 cm

F

7 cm

C

B

5 cm

B 8 cm

5 cm

BD2 = AB2 – AD2

DJ = 6 cm AJ 2 = AD2 + DJ 2

102 – 52

= 75 75 cm cm

AJ =

82 + 62 = 10 cm

VB DB

tan x = ––––

EJ AJ

tan x = ––––

9 = –––– 75 x = 46°6′

7 = ––– 10 x = 35°

The angle between line VD and the base ABC is is ∠VDB and ∠VDB = 46°6′.

©


10 cm

A

A

BD =

x

D

10 cm

D

7 cm

C

m 6 c J

x

G

The angle between line EA and the base ABCD is 35°.

8

Mathematics

7.

Q

9.

R

U

6 cm

M

x

N

SPM

Chapter 20

T

5 cm

P

x

N

S

3 cm

R

T L

9 cm

2

2

PT = PN + NT PT =

4 cm

6 cm

M

P

PS NS

tan x = ––––

2

4 = –– 6 x = 33°41′

92 + 32

90 cm = 90 cm RT 2 = RN 2 – TN 2 RT =

2

Q

The angle between plane PMN and plane RSUT is ∠PNS and and ∠PNS = = 33°41′.

2

5 – 3 = 4 cm RT PT

10.

tan x = ––––

V

4 = –––– 90

7 cm

D

x = 22°52′

x

C

8 cm

6 cm

The angle between line RP and the base LMNP is A

22°52′.

B

VC DC

tan x = –––– 8. E

24 cm

x F D

A

7 = –– 8 x = 41°11′

H 7 cm

C

G

The angle between plane VAD and the base ABCD is ∠VDC and and ∠VDC = = 41°11′.

B

11.

EH HC

tan x = –––– 24 = ––– 7

H

6 cm

G

F

E

x = 73°44′

The angle between line HC and and plane BCEF is is

N

9 cm

∠ ECH

D

C

x

and ∠ ECH = = 73°44′.

8 cm

A

M

B

GN MN

tan x = ––––

9 =— 8 x = 48°22′ The angle between plane ABG and the base ABCD is 48°22′.

9

©


Mathematics

SPM

Chapter 20

12.

14. J

V

M L

K x 12 cm

15 cm

D

H

x

6 cm

N E

N

G

C

5 cm

O

A

M

B

F

8 cm

OM VO

tan x = ––––

HF 2 = HE 2 + EF 2

5 = ––– 12 x = 22°37′

HF =

62 + 82 = 10 cm

10 HN = –––

∠ MVN =

2 = 5 cm

2 x = 45°14′

The angle between plane VAD and plane VBC is and ∠ MVN = = 45°14′. ∠ MVN and

JH HN

tan x = –––– 15 = ––– 5 x = 71°34′

15.

J

The angle between plane JEG and the base EFGH is 71°34′.

M

L

N

K

13 cm

x H

13.

G

D

13 cm

O E

A x

E

HF 2 = HE 2 + EF 2

9 cm

HF = 132 + 132

M

15 cm

338 2

B

OH = –––––

AM 2 = AB2 – BM 2

= 9.192 cm

AM =

152 – 92 = 12 cm

13 NH = –––

2 = 6.5 cm

BM AM

tan x = ––––

NH OH

9 = ––– 12 x = 36°52′

tan x = –––– 6.5 = –––––– 9.192 x = 35°16′

∠ BAE =

2 x = 73°44′

The angle between plane NEG and the base EFGH is 35°16′.

The angle between plane ABD and plane AED is ∠ BAE and and ∠ BAE = = 73°44′.


F

= 338 cm

9 cm

©

13 cm

10

Mathematics

16.

R

U

5 cm

2 = 6 cm

N

tan x = ––––

P x L

12 cm

Chapter 20

12 OT = –––

Q

O

SPM

T

UO OT

4 cm

3 =— 6 x = 26°34′

M

QT 2 = QN 2 – TN 2

The angle between plane UMN and the base LMNP is 26°34′.

QT =

52 – 42 = 3 cm

UO = QT

= 3 cm

11

©


Lines and Planes in 3d

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