THREE DIMENSIONAL SPACE
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THREE DIMENSIONAL SPACE (Angle) By : Drs. Amirullah SMAN 1 kendari PowerPoint PPT Presentation


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THREE DIMENSIONAL SPACE (Angle) By : Drs. Amirullah SMAN 1 kendari. Drs. Arofiq. SMA 5 Mtr. Basic Competence : Determining the central angle between line and plane , and between two planes in space of three dimensional. Drs. Arofiq. SMA 5 Mtr. Material : Angle between two lines

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THREE DIMENSIONAL SPACE (Angle) By : Drs. Amirullah SMAN 1 kendari

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Three dimensional space angle by drs amirullah sman 1 kendari

THREE DIMENSIONAL SPACE

(Angle)

By : Drs. Amirullah

SMAN 1 kendari

Drs. Arofiq

SMA 5 Mtr


Three dimensional space angle by drs amirullah sman 1 kendari

Basic Competence :

Determining the central angle between line and plane, and between two planes in space of three dimensional

Drs. Arofiq

SMA 5 Mtr


Three dimensional space angle by drs amirullah sman 1 kendari

Material :

Angle between two lines

Angle between line and plane

Drs. Arofiq

SMA 5 Mtr


Three dimensional space angle by drs amirullah sman 1 kendari

Angle between

Two Line

Angle between two line is smallest angle that made by two line

m

k

Drs. Arofiq

SMA 5 Mtr


Three dimensional space angle by drs amirullah sman 1 kendari

H

G

E

F

D

C

A

B

Contoh

Diketahui

kubus ABCD.EFGH

Besar sudut antara

garis-garis:

a. AB dengan BG

b. AH dengan AF

c. BE dengan DF

Drs. Arofiq

SMA 5 Mtr


Three dimensional space angle by drs amirullah sman 1 kendari

H

G

E

F

D

C

A

B

Pembahasan

Besar sudut antara

garis-garis:

a. AB dengan BG

= 900

b. AH dengan AF

= 600 (∆ AFH smss)

c. BE dengan DF

= 900 (BE  DF)

Drs. Arofiq

SMA 5 Mtr


Three dimensional space angle by drs amirullah sman 1 kendari

P

Q

V

Sudut antara

Garis dan Bidang

Sudut antara

garis a dan bidang 

dilambangkan (a,)

adalah sudut antara

garis a dan

proyeksinya pada .

Sudut antara garis PQ dengan V

= sudut antara PQ dengan P’Q

= PQP’

P’

Drs. Arofiq

SMA 5 Mtr


Three dimensional space angle by drs amirullah sman 1 kendari

H

G

E

F

D

C

A

B

Contoh 1

Diketahui

kubus ABCD.EFGH

panjang rusuk 6 cm.

Gambarlah sudut

antara garis BG

dengan ACGE,

6 cm

Kemudian hitunglah besar sudutnya!

Drs. Arofiq

SMA 5 Mtr


Three dimensional space angle by drs amirullah sman 1 kendari

H

G

E

F

D

C

A

B

Pembahasan

Proyeksi garis BG

pada bidang ACGE

adalah garis KG

(K = titik potong

AC dan BD)

K

6 cm

Jadi (BG,ACGE) = (BG,KG)

= BGK

Drs. Arofiq

SMA 5 Mtr


Three dimensional space angle by drs amirullah sman 1 kendari

H

G

E

F

D

C

A

B

Pembahasan

BG = 6√2 cm

BK = ½BD

= ½.6√2

= 3√2 cm

∆BKG siku-siku di K

K

6 cm

sinBGK =

Jadi, besar BGK = 300

Drs. Arofiq

SMA 5 Mtr


Three dimensional space angle by drs amirullah sman 1 kendari

H

G

E

F

D

C

A

B

Contoh 2

Diketahui

kubus ABCD.EFGH

panjang rusuk 8 cm.

8 cm

Nilai tangens sudut antara garis CG

dan bidang AFH adalah….

Drs. Arofiq

SMA 5 Mtr


Three dimensional space angle by drs amirullah sman 1 kendari

H

G

E

F

D

C

A

B

Pembahasan

tan(CG,AFH)

= tan (PQ,AP)

= tan APQ

=

=

P

Q

8 cm

Nilai tangens sudut antara garis CG

dan bidang AFH adalah ½√2

Drs. Arofiq

SMA 5 Mtr


Three dimensional space angle by drs amirullah sman 1 kendari

T

a cm

D

C

A

B

a cm

Contoh 3

Pada limas

segiempat beraturan

T.ABCD yang semua

rusuknya sama panjang,

sudut antara TA dan bidang ABCD

adalah….

Drs. Arofiq

SMA 5 Mtr


Three dimensional space angle by drs amirullah sman 1 kendari

T

a cm

D

C

A

B

a cm

Pembahasan

• TA = TB = a cm

• AC = a√2 (diagonal

persegi)

•∆TAC = ∆ siku-siku

samakaki

sudut antara TA dan bidang ABCD

adalah sudut antara TA dan AC

yang besarnya 450

Drs. Arofiq

SMA 5 Mtr


Three dimensional space angle by drs amirullah sman 1 kendari

Sudut antara

Bidang dan Bidang

Sudut antara

bidang  dan bidang 

adalah sudut antara

garis g dan h, dimana

g  (,) dan h  (,).

(,) garis potong bidang  dan 

h

(,)

g

Drs. Arofiq

SMA 5 Mtr


Three dimensional space angle by drs amirullah sman 1 kendari

H

G

E

F

D

C

A

B

Contoh 1

Diketahui kubus

ABCD.EFGH

a. Gambarlah sudut

antara bidang BDG

dengan ABCD

b. Tentukan nilai sinus

sudut antara BDG

dan ABCD!

Drs. Arofiq

SMA 5 Mtr


Three dimensional space angle by drs amirullah sman 1 kendari

H

G

E

F

D

C

A

B

Pembahasan

a. (BDG,ABCD)

• garis potong BDG

dan ABCD  BD

• garis pada ABCD

yang  BD  AC

• garis pada BDG

yang  BD  GP

P

Jadi (BDG,ABCD) = (GP,PC)

=GPC

Drs. Arofiq

SMA 5 Mtr


Three dimensional space angle by drs amirullah sman 1 kendari

H

G

E

F

D

C

A

B

Pembahasan

b. sin(BDG,ABCD)

= sin GPC

=

=

= ⅓√6

P

Jadi, sin(BDG,ABCD) = ⅓√6

Drs. Arofiq

SMA 5 Mtr


Three dimensional space angle by drs amirullah sman 1 kendari

T

9 cm

A

C

6 cm

B

Contoh 2

Limas beraturan

T.ABC, panjang

rusuk alas 6 cm dan

panjang rusuk tegak

9 cm. Nilai sinus sudut

antara bidang TAB

dengan bidang ABC

adalah….

Drs. Arofiq

SMA 5 Mtr


Three dimensional space angle by drs amirullah sman 1 kendari

T

9 cm

A

C

6 cm

B

Pembahasan

•sin(TAB,ABC)

= sin(TP,PC)

= sinTPC

•TC = 9 cm, BP = 3 cm

•PC =

=

•PT =

=

P

3

Drs. Arofiq

SMA 5 Mtr


Three dimensional space angle by drs amirullah sman 1 kendari

• Lihat ∆ TPC

PT = 6√2, PC = 3√3

Aturan cosinus

TC2 = TP2 + PC2 – 2TP.TC.cosTPC

81 = 72 + 27 – 2.6√2.3√3.cosTPC

36√6.cosTPC = 99 – 81

36√6.cosTPC = 18

cosTPC =

=

T

9 cm

6√2

A

C

2

1

3√3

P

B

Drs. Arofiq

SMA 5 Mtr


Three dimensional space angle by drs amirullah sman 1 kendari

• Lihat ∆ TPC

cosP =

Maka diperoleh

Sin P =

Jadi sinus (TAB,ABC)

=

12

P

√6

Drs. Arofiq

SMA 5 Mtr


Three dimensional space angle by drs amirullah sman 1 kendari

H

G

E

F

D

C

A

B

Contoh 3

Diketahui kubus

ABCD.EFGH, pan-

jang rusuk 4 cm

Titik P dan Q

berturut-turut

di tengah-tengah

AB dan AD.

4 cm

Q

P

Sudut antara bidang FHQP dan bi-

dang AFH adalah . Nilai cos =…

Drs. Arofiq

SMA 5 Mtr


Three dimensional space angle by drs amirullah sman 1 kendari

H

G

E

F

D

C

A

B

Pembahasan

• (FHQP,AFH)

= (KL,KA)

= AKL = 

• AK = ½a√6 = 2√6

• AL = LM = ¼ AC

= ¼a√2 = √2

• KL =

=

=3√2

4 cm

K

Q

L

M

P

Drs. Arofiq

SMA 5 Mtr


Three dimensional space angle by drs amirullah sman 1 kendari

Pembahasan

• AK = 2√6 , AL = √2

KL = 3√2

Aturan Cosinus:

AL2 = AK2 + KL2 – 2AK.KLcos

2 = 24 + 18 – 2.2√6.3√2.cos

24√3.cos = 42 – 2

24√3.cos = 40

cos =

K

M

L

A

Jadi nilai cos =

Drs. Arofiq

SMA 5 Mtr


Three dimensional space angle by drs amirullah sman 1 kendari

“Sesungguhnya bersama kesulitan ada kemudahan”

(Al-Insyirah : 6)

Selamat Belajar, semoga sukses !

Drs. Arofiq

SMA 5 Mtr


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