ปีทาโกรัส2

ทฤษฎีบทปีทาโกรัส ระดับชั้นม.2 ครูบุญยอด นันตะสุข

1 1. 3.2 5. 2, 1) 3.2 6.1 4. 3.2 2) 1. 2. 6.1

2) 2. 2) 3.2 3. 2) 3.2 3.2 2) 1)

2 (Pythagoras’ theorem) ABC AC^ B A AB AC CB 1. 3 . 4

B C BC 4 3. 5 52 ϭ 32ϩ42

2. 6 8. 3. 2.4 1, 2, 3 ABC AC c ab c Ba

3 8 6. 10 102 ϭ 62ϩ82 3.2 4 42 ϭ 3.22ϩ2.42 3.2 . CB 2.4 . A b C c2 ϭ a2ϩb2

4 1. 2. a z y 4. x 5. 1.5 m 3.6 (1) a2 ϭ 92ϩ ϭ 81ϩ a 12 ϭ 225 9 ϭ 15ϫ (1) a ϭ 15 a a ϭ 15 15

3. 10 a b 6 12 2.4 n 2.6 (2) 10 a 26 ϩ122 ϩ144 5 ϫ15 (Ϫ15)ϫ(Ϫ15) Ϫ15

(2) 262 ϭ a2ϩ a2 ϭ 262 ϭ 676 ϭ 576 ϭ 24ϫ a ϭ 24 a a ϭ 24 24 1. x x x 29 20 16 12 x 2. A B DC D C 3. 25 BD ϭ A 15 B

5 ϩ102 2Ϫ102 6Ϫ100 6 ϫ24 (Ϫ24)ϫ(Ϫ24) Ϫ24 x 3.5 1.2 34 30 x ⌬ABC BC ϭ 18 AB ϭ AC ϭ 15 ⌬ABC AD BC ᮀABCD AB ϭ 15 ϭ 25 ᮀABCD

6 A BC 1. 3 2. 1 ᮀARQB AB AC , CB NA M C O

⌬ABC BC ABC AC AB 3 ABC AC CB B AB ,ᮀACMN ᮀCBPO R Q B P

3. AM , NC , CP OB 4. ᮀACMN ᮀCBPO N A 5 4 1 7 M 2 3 C 8 O 5. 8 1. 2 CB AC 2. ᮀBCHI AB , AC G A E D

4 7 R ᮀARQB Q 5B 6 7 P ᮀARQB ABC ACB BC ,ᮀABDE ᮀAGFC F CH I B

8 3. ᮀAGFC ᮀBCHI HI BC G 1 A E 4. D 1, 2, 3 ᮀABDE 5( 12 R 5 A8 4 1 N 26 Q 3 MC 7 B OP 1

ᮀBCHI 4 JK F H 2 C 3 J 4 K 5 I B 3, 4 5) ᮀABDE 12 GF A C H 2 I E 15 3 4B D 2

1. 3 BC AC 2. ᮀBCHI AB , AC F C G A E 3. ᮀBCHI 4. O j jj k F C G A 5. ᮀAGFC E ᮀABDE 4

9 ABC ACB BC ,ᮀABDE ᮀAGFC H I B D AB O 4 k ᮀBCHI H OI k B j D ᮀABDE ᮀABDE ᮀBCHI

10 c c a c c2 ϭ 1 ab ,(1) a ϭ 12 b ϭ 5 ,(3) a ϭ 7 b ϭ 24 (1) c2 ϭ a2ϩ a ϭ 12 bϭ5 c2 ϭ 122ϩ52 ϭ 144ϩ25 ϭ 169 ϭ 13ϫ13 c ϭ 13 13

a2 b2 ab c2 b a ab a2ϩb2 c ,(2) a ϭ 1.6 b ϭ 3 ϩb2

(2) c2 ϭ a2ϩ a ϭ 1.6 bϭ3 c2 ϭ 1.62ϩ32 ϭ 2.56ϩ9 ϭ 11.56 ϭ 3.4ϫ3.4 c ϭ 3.4 3.4 c2 ϭ a2ϩ (3) aϭ7 b ϭ 24 c2 ϭ 72ϩ242 ϭ 49ϩ576 ϭ 625 ϭ 25ϫ25 c ϭ 25 25 ab a, b c c2 ϭ a2ϩ ca b2 ϭ c2Ϫ cb a2 ϭ c2Ϫ

11 ϩb2 ϩb2 c ϩb2 ..........(1) b (1) Ϫa2 ..........(2) a (1) Ϫb2 ..........(3)

12 2 c ,(1) b ϭ 15 c ϭ 39 ,(3) b ϭ 4.5 c ϭ 7.5 (1) a2 ϭ c2Ϫb2 b ϭ 15 c ϭ 39 a2 ϭ 392Ϫ152 ϭ ϭ 1,521Ϫ2 1,296 ϭ 36ϫ36 a ϭ 36 36 b2 ϭ c2Ϫa2 (2) a ϭ 24 c ϭ 30 b2 ϭ 302Ϫ242 ϭ 900Ϫ576 ϭ 324 ϭ 18ϫ18 b ϭ 18 18 a2 ϭ c2Ϫb2 (3) b ϭ 4.5 c ϭ 7.5 a2 ϭ 7.52Ϫ4.5 ϭ 56.25Ϫ2 ϭ 36 ϭ 6ϫ6 aϭ 6 6

ab ,(2) a ϭ 24 c ϭ 30 2 225 2 6 52 20.25

3 10 D 10 . . B 10 C 14 . E A DC C AE ABCE ADE A AD2 ϭ AE AE ϭ BC DE ϭ DC ϭ 10ϩ ϭ 24 AD2 ϭ 102 ϭ 100 ϭ 676 ϭ 26ϫ AD ϭ 26 2 4 CD ϭ 17 BC ϭ 13 13 B 12 . A

13 14 10 A E AB 14 BC 10 CD 10 AD BC AED E2ϩDE2 C ϭ 10 CϩCE ϩ14 (CE ϭ AB) 2ϩ242 0ϩ576 6 ϫ26 26 ABCD AB ϭ 12 .C E 17 . D

14 BE AD CD ⌬BCE EC ϭ CD ⌬BCE ϭ 17Ϫ ϭ5 BC2 ϭ BE 132 ϭ BE BE2 ϭ 132 ϭ 169 ϭ 144 ϭ 12ϫ BE ϭ 12 AB 174 ab ,1. a ϭ 8 b ϭ 15 ,3. b ϭ 2.1 c ϭ 2.9

E ᮀABED DϪED Ϫ12 (ED ϭ AB) E2ϩCE2 E2ϩ52 2Ϫ52 9Ϫ25 4 ϫ12 BCD ϭ 1 ϫ(12ϩ17)ϫ12 2 ϭ 29ϫ6 ϭ 174 c ,2. a ϭ 12 b ϭ 35 ,4. a ϭ 2.4 c ϭ 3

1. c cc (1) 3 25 (3) 4 (5) c 60 1.2 0.9 c (7) c 6.3 1.6 (9) c 28 53

15 c 1c 33 (2) 2.4 (4) c (6) 56 (8) 60 c 45 c5 (10) 1.4 cb 41 40

16 aa 2. 17 (1) 12 (2) 2 20 (3) 2 3. 50 a 16 18 172 ϭ 152 2 3 90

a 22 a 25 7 4 8 48 15 17 1 2ϩ82

34 5 52 ϭ 32ϩ42 Z 54 Y 3X ⌬XYZ XYZ x, y A c Ba ⌬ABC AB ϭ c BC ⌬ABC EF DE a ⌬DEF Х ⌬ABC

17 ⌬XYZ XY ϭ 3 XZ ϭ 4 YZ ϭ 5 YXZ 90Њ yz x2 ϭ y2ϩz2 x b aC Cϭa AC ϭ b b2 ϭ a2ϩc2 ABC DEF DEF c

18 DEF ac D c Ea EF ϭ BC ϭ a DE ϭ AB ϭ ⌬DEF DF2 ϭ a2ϩc2 ⌬ABC b2 ϭ a2ϩc2 DF2 ϭ b2 DF ϭ b ⌬DEF Х ⌬ABC DE^F ϭ AB^ C ϭ ⌬ABC 15 1 ⌬ABC . B 39 15 . A 36 . C c2 ϭ a2ϩb2 ⌬ABC

F DEF EF DE aF ( ) ϭc () () () ϭ 90Њ ( ) ( ) ABC 39 ⌬ABC 36 a ϭ 15 b ϭ 36 c ϭ 39 a2 ϭ 225 ,b2 ϭ 1 296 ,c2 ϭ 1 521 ,a2ϩb2 ϭ 225ϩ1 296 ϭ 1,521

2 ABC ⌬ABD B ⌬ACD 16 ⌬ABC AB2 ϭ AD ϭ 122 ϭ 144 AB2 ϭ 400 AC2 ϭ AD ϭ 122 ϭ 144 AC2 ϭ 225 AB2ϩAC2 ϭ 400 ϭ 625 BC2 ϭ (BD ϭ (16 ϭ 252 ϭ 625 BC2 ϭ AB

19 ⌬ABC A 12 C 9 D D2ϩBD2 2ϩ162 4ϩ256 0 D2ϩCD2 2ϩ92 4ϩ81 5 0ϩ225 5 DϩDC)2 6ϩ9)2 2 5 B2ϩAC2 BAC

20 1. 2 (2) (1) 1.2 5 (4) 1.6 1 (5) 65 16 5 63 2. 12 (1) 9 (2) (3) 15 (4) (5)

14 (3) 37 13 35 12 12.5 15 (6) 4 9.5 10.4 9 12 18 13 12 15 26 10 16 12

3. 7 (1) (2) 30 (3) (4) 6 (5) (6) 12 (7) (8) C A (9) BC AB 80 64 B A 80 (10) 12 9

21 7 24 34 16 28 13 24 20 B AC BC AB 64 . C 0. 9 12

22 4. 12 5. ABCDEFGH DH 8 H 8. E D 9. A 12

14 23 AB 12 AD 9 BH G F C .B

ab 1. 2. 3. a2 4. a, b, c (na)2ϩ(nb) a2ϩb2 ϭ c2

23 c a2ϩb2 ϭ c2 1 a2ϩb2 ϭ c2 12 2ϩb2 ϭ c2 nϾ0 )2 ϭ (nc)2

24 1. (2) (1) 5.6 4.8 5 (4) 48 72 20 168 (6) a c 21 2. ac

(3) 45 6.5 27 (5) 51 24 (7) 4 13 12 7.

3. (1) 48 14 . (2) 12 13 4. a, b, c a ϭ 3, b ϭ 4, c ϭ 3 ,3a ϭ 9 3b ϭ 12 3 152 ϭ 92ϩ122 225 ϭ 81ϩ144