Tugas Akhir Mata Kuliah Geometri Transformasi FMIPA Unnes Rombel 03 Kelompok 7 Akhmad Faid M. (4101415012) Rena Yuliantika S. (4101415028) Fauzi Badar B. (4101415050) Ragil Yunitasari (4101415087) Melati Eka P. (4101415125) Info: faidmurtadlo@students.unnes.ac.id

2. Tugas Akhir Geometri Transformasi
Dosen Pengampu: Drs. Suhito, M.Pd.

3. Oleh Kelompok 7
1. Akhmad Faid Murtadlo 4101415012
2. Rena Yuliantika Shora 4101415028
3. Fauzi Badar Bagaskara 4101415050
4. Melati Eka Putri 4101415125
5. Ragil Yunitasari 4101415087

4. Diketahui :
1. 𝐾𝐿 sembarang,
2. Garis t sembarang,
3. 𝑃𝑄 ∕∕ 𝑡
4. 𝑒 ⊥ 𝑓, 𝑒 ∩ 𝑓 = {𝐵}
5. 𝑔 ∩ ℎ = {𝐴}, 𝑚 ∠𝑔, ℎ =
1
2
𝜑1

5. K
L
P
Q
B
A
1
2
𝜑1
Diketahui

6. 𝑅 𝐴,𝜑1
∘ 𝑀𝑠 ∘ 𝑆 𝐵 ∘ 𝐺 𝑃𝑄 ∘ 𝑀𝑡 ∘ 𝐺 𝐾𝐿 (△ 𝑋𝑌𝑍)
= 𝑅 𝐴,𝜑1
∘ 𝑀𝑠 ∘ 𝑆 𝐵 ∘ 𝐺 𝑃𝑄 ∘ 𝑀𝑡 (△ 𝑋1 𝑌1 𝑍1)
= 𝑅 𝐴,𝜑1
∘ 𝑀𝑠 ∘ 𝑆 𝐵 (△ 𝑋3 𝑌3 𝑍3)
= 𝑅 𝐴,𝜑1
∘ 𝑀𝑠 (△ 𝑋4 𝑌4 𝑍4)
= 𝑅 𝐴,𝜑1
(△ 𝑋5 𝑌5 𝑍5)
=△ 𝑋6 𝑌6 𝑍6
Bukti Langsung

7. GAMBAR BUKTI LANGSUNG

8. K
L
𝑋
𝑌
𝑍
𝑋1
𝑌1
𝑍1
𝑋2
𝑌2
𝑍2
P
Q
𝑋3
𝑌3 𝑍3
B
𝑍5
𝑌5
𝑋5
𝑍4
𝑌4
𝑋4
A
𝑍6
𝑌6
𝑋6
1
2
𝜑1

9. Diketahui :
1. 𝑎// 𝑏, 𝑎 ⊥ 𝐾𝐿, 𝑑 𝑎,𝑏 =
1
2
𝐾𝐿
2. 𝑐//𝑑, 𝑐 ⊥ 𝑃𝑄, 𝑡//𝑃𝑄, 𝑑 𝑐,𝑑 =
1
2
𝑃𝑄
3. 𝑒 ⊥ 𝑓, 𝑒 ∩ 𝑓 = {𝐵}
4. 𝑔 ∩ ℎ = {𝐴}, 𝑚 ∠𝑔, ℎ =
1
2
𝜑1

10. K
L
𝑎 𝑏
P
Q
f
B
A
1
2
𝜑1
Diketahui

11. Bukti Tak Langsung
𝑅 𝐴,𝜑1
∘ 𝑀𝑠 ∘ 𝑆 𝐵 ∘ 𝐺 𝑃𝑄 ∘ 𝑀𝑡 ∘ 𝐺 𝐾𝐿
= 𝑀ℎ ∘ 𝑀𝑔 ∘ 𝑀𝑠 ∘ 𝑀𝑓 ∘ 𝑀𝑒 ∘ 𝑀 𝑑 ∘ 𝑀𝑐 ∘ 𝑀𝑡 ∘ 𝑀 𝑏 ∘ 𝑀 𝑎 __(1)
= 𝑅 𝐴,𝜑1
∘ 𝑀𝑠 ∘ 𝑀𝑓 ∘ 𝑀𝑒 ∘ 𝑀 𝑑 ∘ 𝑀𝑐 ∘ 𝑀𝑡 ∘ 𝑀 𝑏 ∘ 𝑀 𝑎______(2)
= 𝑅 𝐴,𝜑1
∘ 𝑅 𝐶,𝜑2
∘ 𝑅 𝐷,𝜑3
∘ 𝑀𝑐 ∘ 𝑅 𝐸,𝜑4
∘ 𝑀 𝑎_______________(2)
= 𝑅 𝐴,𝜑1
∘ 𝑀𝑖 ∘ 𝑀𝑗 ∘ 𝑀𝑗 ∘ 𝑀 𝑘 ∘ 𝑀𝑐 ∘ 𝑅 𝐸,𝜑4
∘ 𝑀 𝑎_________(3)
= 𝑅 𝐴,𝜑1
∘ 𝑀𝑖 ∘ 𝐼 ∘ 𝑀 𝑘 ∘ 𝑀𝑐 ∘ 𝑅 𝐸,𝜑4
∘ 𝑀 𝑎________________(3)
= 𝑅 𝐴,𝜑1
∘ 𝑀𝑖 ∘ 𝑀 𝑘 ∘ 𝑀𝑐 ∘ 𝑅 𝐸,𝜑4
∘ 𝑀 𝑎_________________(4)
= 𝑅 𝐴,𝜑1
∘ 𝑀𝑖 ∘ 𝑅 𝐺,𝜑5
∘ 𝑅 𝐸,𝜑4
∘ 𝑀 𝑎____________________(4)
= 𝑅 𝐴,𝜑1
∘ 𝑀𝑖 ∘ 𝑀𝑙 ∘ 𝑀 𝑚 ∘ 𝑀 𝑚 ∘ 𝑀 𝑛 ∘ 𝑀 𝑎_____________(5)
= 𝑅 𝐴,𝜑1
∘ 𝑀𝑖 ∘ 𝑀𝑙 ∘ 𝐼 ∘ 𝑀 𝑛 ∘ 𝑀 𝑎_____________________(5)
= 𝑅 𝐴,𝜑1
∘ 𝑀𝑖 ∘ 𝑀𝑙 ∘ 𝑀 𝑛 ∘ 𝑀 𝑎 ____________________(6)
= 𝑅 𝐴,𝜑1
∘ 𝑅 𝐻,𝜑6
∘ 𝑅𝐽,𝜑7
__________________________(6)
= 𝑅 𝐴,𝜑1
∘ 𝑀 𝑜 ∘ 𝑀 𝑝 ∘ 𝑀 𝑝 ∘ 𝑀 𝑞 ____(7)
= 𝑅 𝐴,𝜑1
∘ 𝑀 𝑜 ∘ 𝐼 ∘ 𝑀 𝑞____________(7)
= 𝑅 𝐴,𝜑1
∘ 𝑀 𝑜 ∘ 𝑀 𝑞 ____________(8)
= 𝑅 𝐴,𝜑1
∘ 𝑅 𝑀,𝜑8
________________(8)
= 𝑀𝑟 ∘ 𝑀 𝑢 ∘ 𝑀 𝑢 ∘ 𝑀𝑣 _________(9)
= 𝑀𝑟 ∘ 𝐼 ∘ 𝑀𝑣________________(9)
= 𝑀𝑟 ∘ 𝑀𝑣_________________(10)
= 𝑅 𝑁,𝜑9
___________________(10)

12. Syarat pada Bukti Tak Langsung
1) 𝑎// 𝑏, 𝑎 ⊥ 𝐾𝐿, 𝑑 𝑎,𝑏 =
1
2
𝐾𝐿
 𝑐//𝑑, 𝑐 ⊥ 𝑃𝑄, 𝑡//𝑃𝑄, 𝑑 𝑐,𝑑 =
1
2
𝑃𝑄
 𝑒 ⊥ 𝑓, 𝑒 ∩ 𝑓 = {𝐵}
 𝑔 ∩ ℎ = {𝐴}, 𝑚 ∠𝑔, ℎ =
1
2
𝜑1
2) 𝑏 ∩ 𝑡 = {𝐸}, 𝑚 ∠𝑏, 𝑡 =
1
2
𝜑4
 𝑑 ∩ 𝑒 = 𝐷 , 𝑚 ∠𝑑, 𝑒 =
1
2
𝜑3
 𝑓 ∩ 𝑠 = {𝐶}, 𝑚 ∠𝑏, 𝑡 =
1
2
𝜑2
3) 𝑗 ∩ 𝑖 = {𝐶}, 𝑚 ∠𝑗, 𝑖 =
1
2
𝜑2
 𝑘 ∩ 𝑗 = 𝐷 , 𝑚 ∠𝑘, 𝑗 =
1
2
𝜑3
4) 𝑐 ∩ 𝑘 = 𝐺 , 𝑚 ∠𝑐, 𝑘 =
1
2
𝜑5
5) 𝑚 ∩ 𝑙 = {𝐺}, 𝑚 ∠𝑚, 𝑙 =
1
2
𝜑5
 𝑛 ∩ 𝑚 = 𝐸 , 𝑚 ∠𝑚, 𝑛 =
1
2
𝜑4
6) 𝑙 ∩ 𝑖 = {𝐻}, 𝑚 ∠𝑙, 𝑖 =
1
2
𝜑6
 𝑎 ∩ 𝑛 = 𝐽 , 𝑚 ∠𝑎, 𝑛 =
1
2
𝜑7
7) 𝑝 ∩ 𝑜 = {𝐻}, 𝑚 ∠𝑝, 𝑜 =
1
2
𝜑6
 𝑞 ∩ 𝑝 = 𝐽 , 𝑚 ∠𝑞, 𝑝 =
1
2
𝜑7
8) 𝑞 ∩ 𝑜 = 𝑀 , 𝑚 ∠𝑞, 𝑜 =
1
2
𝜑8
• g ∩ ℎ = 𝐴 , 𝑚 ∠𝑔, ℎ =
1
2
𝜑1
9) 𝑢 ∩ 𝑟 = {𝐴}, 𝑚 ∠𝑢, 𝑟 =
1
2
𝜑1
 𝑣 ∩ 𝑢 = 𝑀, 𝑚 ∠𝑣, 𝑢 =
1
2
𝜑8
8) 𝑣 ∩ 𝑟 = {𝑁}, 𝑚 ∠𝑣, 𝑟 =
1
2
𝜑9

13. GAMBAR BUKTI TAK LANGSUNG

14. K
L
𝑋
𝑌
𝑍
𝑋1
𝑌1
𝑍1
𝑎 𝑏
𝑋2
𝑌2
𝑍2
P
Q
𝑋3
𝑌3 𝑍3
f
B
𝑍5
𝑌5
𝑋5
𝑍4
𝑌4
𝑋4
A
𝑍6
𝑌6
𝑋6
1
2
𝜑1

15. 1
2
𝜑5
K
L
𝑋
𝑌
𝑍
𝑎 𝑏
P
Q
f
B
A
𝑍6
𝑌6
𝑋6
E
D
C
i
G
JH
o
1
2
𝜑7
M
1
2
𝜑1
N
Proses
Bukti Tak
Langsung

16. TERIMA KASIH