teorema sacheri legendre

1.      Lemma: Jumlah besar sudt dalam setiap segitiga kurang dari 180

Adib: s CAB + CBA  <180

Jawab: s ABC + ABD =180

berdasarkan teorema sudut luar , s ABD > s CAB

jadi,

sABC+ ABD =180

s ABD =180- s ABC

shg 180- s ABC > CAB

s CAB + ABC <180 (terbukti)

2.      Lemma: Diketahui segitiga ABC maka ada A1B1C1 sehingga sudut A1 +B!+C1 = A+B+C dan A1 <= ½ A

Ada segitiga ABC

E adalah titik tengah BC Sehingga BE=EC

Melalui titik A dan E dapat dilukis garis AE, perpanjang AE sehingga terdapat titik F diman AE=EF

Lukis garis CF dan BF

Karena s AEB=CEF maka segitiga ABE = FCE (S.SD.S)

s A+B+C = s 1+ 2 +3 + 4

= s 1 + 21+31+4

= s CAF + AFC + FCA

Selanjutnya : s A = s 1+s2

                      s A = s1+s21

tidak mungkin sudut 21 > ½ CBA dan s1 > ½ s CAB

jadi: sudut 21 >= ½ CBA dan s1 >= ½ s CAB

sehingga segitiga A1B1C1 = ABC dan A1<= ½ A (terbukti)

           

3.      Jumlah besar sudut  dalam segitiga adalah kurang sama dengan 180’

Andaikan diketahui segitiga ABC dan andaikan s A + B+ C > 180, maka ada P>) sehingga s A+B+C=180+P

Dengan  (Lemma: Diketahui segitiga ABC maka ada A1B1C1 sehingga sudut A1 +B!+C1 = A+B+C dan A1 <= ½ A) jadi ada segitiga A1B1C1 sehingga sA1+B1+C1= 180+P dan s A1 <= A

Ada segitiga  A2B2C2 sehingga sA2+B2+C2= 180+P dan s A2 <= A

Jika titik ulangi lagi, maka terdapat satu barisan segitiga –segitga  A1B1C1, A2B2C2, ..., AnBnCn, masing2 dengan jumlah sudut 180+P, sedemikian sehingga untuk sembarang n>0, s An <= 1/2n <A<=P

Jadi s An + Bn + Cn <P+Bn+ Cn <= P Bn + Cn

Sehingga  s Bn + Cn >= 180

Hal ini bertantangan dengan lemma (Lemma: Jumlah besar sudt dalam setiap segitiga kurang dari 180)

Jadi pengandaian bahwa ada segitiga ABC dg s A+ B+C >180 adalah tidak benar. Haruslah s A+B+C<=180 (terbukti)

4.      Corollery: jumlah sudut pada segi emmpat yang convex adalah kurang dari atau sama dengan 380

Bukti: pandang segitiga ABD

Berdasarkan teoorema segi empat sucheri maka

S A+ B+C =<= 180 .....1

Panadang segitia BCD Berdasarkan teoorema segi empat sucheri maka

S B+C +D<= 180 ......2

Akibat pers 1 dan 2 maka S A+B+C+D<=360

5.      Bukti bahwa sudut eksterior suatu segitiga lebih besar atau sama dengan jumlah ukuran dua sudut interiopr didepannya

Bukti;

Diketahuio segitiga PQR dan Sudut PRS adalah sudut eksterior

Adib

S PRS >= QPR + PQR

Bukti: bverdasarkan sudut lurus, maka:

S PRQ + PRS = 180

Sehingga PRQ + PRS = 180

S PRS + PRQ –PRS= 180-PRS

S PRQ = 180 – PRS...2

Berdasarkan teorema sacheri segiemapt maka:

S PQR + QPR + PRQ <= 180...2

Sehingga subtitusikan 1 dan 2

S PQR + QPR + PRQ <= 180

S PQR + QPR <= 180-PRQ

S PQR + QPR <= 180- (180-PRS)

S PQR + QPR <= PRS

JADi terbukti sudut PRS >=  S PQR + QPR

6.      Jika dua garis dipotong oleh garis transpersal , kedua garis terbut diperpanjnag tak terbatas dan akan diperpanjang dan akan berpotongan maka pasangan sudut dalam jumlahnya kurang dari siku-siku

Bukti:

Misalkan diketahui garis m,n danl

Garsi l memotong n dan di tiik q dan m di titik p. Membentuk pasangan sudut dalam sepihak yaitu s P4 dan Q1

Garis m dan n diperpanjang tak terbatas sehingga berpotongan di titik R oleh karena itu membentuk segitgia PQR

Adib s P4 + Q1 < 180

Bukti: berdasrkan lemma (jumlah 2 sudut segitiga adalah 180) maka P4 + Q1 <180

Jadi terbukti P4 + Q1 <180

7.      Diagonal segiempat acheri adalah kongruen

Misalkan

Diketahui segiemapt ABCD

Tarik Ac dan BD shg membnetuk 2 segitigsa ABD dan BCA

Ad= BC (defnisi)

S A = s B

AB = Ab (refleksi)

Maka segitiga ABD = BCA(s.sd.s) akibaynya Ac=BD

Jdi terbukti bahwa diagonal segiempat scheri adalah kongruen

8.      Sudut-sudut segiempat sacheri adsalah kongruen

Misalkan

Diketahui segiemapt ABCD

Tarik Ac dan BD shg membnetuk 2 segitigsa ACD dan BDC

Ad= BC (defnisi)

AC = BD

DC = DC (refleksi)

Maka segitiga ACD = BDC(s.s.s) akibatnya s D= s C

9.      Sudut-sudut puncak segiempat sacheri adalah tidak tumpul dan keduanya lancip atau keduanya siku-siku

Buktiberdasrkan corollary (jumlah segiemapat convex adalah kurang dari atau sam dengan 360)

S A+B+C+D<= 360

90+90+ C+D <= 360       D=C

180+C+C <= 360

2C <= 180

S C = 90

10.  Garis yang menghubungkan titik tengah dari sisi atas dan sisi alas dari segiempat sacheri adalah tegak lurus terhadap sisi atas maupun sisi bawah

Padang segitiga ADM dan BCM

AD=BC (def), s DAM= s CBM (def) AM=BM (def) maka segitiga ADM=BCM (s.sd.s) akibatnya DM=CM dan DMA = CMB..... 1

Padang seegitiga MDN dan MCN

Dn=Cn (dik), MN=Mn (ref), DM=Cm (akibat segitiga ADM = BCM) maka segitiga MDN = MCN akibatnya s DNM= CNM

Sedangakan a DHA + CMB =1 80 maka s DMA = cmb = 90

Sehg MN tegak lurus DC di N.... 2

Segitiga MDN = MCN juga mengakibatkan s DMN = CMN ...3

Selanjutnya s AMN = DMA + DMN (dr 1 dan 2)

Dg demikian s AMN + BMN

Sedangkan s AMN + BMN = 180 maka AMN = BMN = 90

Sehingga terbukti MN tegak lurus AB di M

11.  Sisi atas dan sisi bawah segiempat sacheri adalah sejajar

Berdasarkan teorema (Garis yang menghubungkan titik tengah dari sisi atas dan sisi alas dari segiempat sacheri adalah tegak lurus terhadap sisi atas maupun sisi bawah) sehingga MN tegak lurus AB dan MN tegak lurus CD maka sisi atas dan als sejajar.

12.  Pada bebrapa segiempat sacheri panjang sisi atas lebih panjang atau sam dengann sis bawah.

Misalkan segiempat sacheri ABCD

Adib: AB<= CD

Bukti:

Lukiskan BD

Kemungkinan terbentuk sudut s 1 = 2, 1 <2, 1>2

Jika s 1 = 2 maka segitiga ADB = CBD dan AB= CD

Jika s 1< 2 maka berdasarkan teorema hinge AB<CD

Jika s 1 > 2 ....1

S 2 + 3 = 90

S2 = 90-s3...2

Dar 1 dan 2 maka:

S1 > s 2

S 1> 90-s3

S 1+ s3 > 90

Sehingga :

S 1 + 3 + BAD > 180

Hal ini kontradiksi dengan etorema sacheri segiempat (jumlah sudut dalam segitiga kurang dari atau sama dengan 180)

Haruslah s 1 <= s 2 shgga AB <= CD

soal bahasa inggris

1. Which one is incorrect?

e. Do you know that 100,000 is read as a hundred thousands?

2. Which one is incorrect?

d. The First Sultan Thaha

3. Which one of the following writing is not acceptable?

e. the 10 of August.

4. To express “a truck which has weight of 10 ton”, we may say

b. A ten-ton truck

5. To express that “you have walked two kilometres” you can write

a. I have a two-kilometre walk   

6. Among the following sentences, there is one with incorrect grammar:

c. Millions ducks died on the beach.

7. The correct use of the word “one” in the following sentences is in

.

d. He needs one pencil and twenty ones-page book.

8. Suppose we want to say ‘3+2=5’. Out of the following, which sentences cannot be used?

a.    Three added by two is the same as five.

9. In conversational style, dealing with small numbers, people can say the following, except

c.  8-4=4   :  Eight taken by four is the same as four.

10. How do you say ‘3 ´ 4 = 12’?

a.    ”.

b.    We may say “three times four equals twelve”.

11. Which one is not acceptable based on grammar?

a. 24 : 3 = 8   :         Twenty four divided by three equal eight.

12. The following is the way we express a fraction in words. The incorrect expression is

b.  :  three over fifth

13. The following collections which can be classified as a set is  …

c. A group of delicious food in a birthday party.

14. Which one is an empty set?

c. A set of prime numbers which are less than 2.

15. Which pair of equal sets in the following information?

    A = {x | x is a letter which constructs the word ”asmarandana”}.

16. Suppose that both x and y are odd numbers. Then

d.   is an even number.

17. Let x be a positive integer and y be a negative integer. Then

b.   is a positive integer.

18. The incorrect statement in the following set of statements is

e.  The set of kites is a subset of the set of squares.

19. Suppose that A is a set of squares, B is a set of rhombuses, C is a set of rectangles. Then

     a.  A is the subset of C.

20. Which one is incorrect?

c.  A trapezoid is a kite.

21. Which one is correct?

e.  A rectangle is a parallelogram and a square is a rhombus.

/

/

22. Let A be a set of squares, B be a set of parallelograms, and C be a set of rhombuses. Then the we cannot conclude that

c. B Ì A.    

23. Let P be a set of squares, Q be a set of rectangles, and R be a set of parallelograms. Then

b.  P Ç Q = P

24. If A, B and C are sets, which statement is incorrect?

e.  If AÌ B, and B Ë C, then A Ç C = Æ.

25. Suppose A and B are the sets of natural numbers. The only correct statement is

c. If A ¹ B, then A Ì B.

26. If A, B, and C are sets, then the incorrect statement is

a. A Ç (A È B ) = B.

27. Let A  be the set of integers. The identity element of A under multiplication operation is

      b. 0

28. Let A  be the set of integers. The inverse of -4 under multiplication operation is

b. .             

29. Let B  be the set of integers. The inverse of -5 under addition operation is

.        d. 5

30. Let B be the set of natural numbers. The identity element of B under addition operation is

  .           e. none.

31. Let K be the set of integers. The inverse of x in the set K under addition operation is

      a. –x

32. Let K be the set of integers. The inverse of x in the set K under multiplication operation is

d. .        e. x.

33. Suppose that a and b are the elements of the set of integers A and a * b = a + b – 2ab. The identity element under this operation is

      b. 0.           c. a.            d. –b.             e. b. 

34. The inverse element of a under the operation defined in question number 33 (above) is

      c. .            d. –.             e. . 

35. The solution set for the equation of | x+1|£ 1, where x is an integer, is

b. { -1}.      c. { 0 }.     d. { 1}.     e. { 2 }.

36.  Suppose we have a statement “Paris is in France”. Then

a. The truth value (nilai kebenaran) of this statement is F (false).

37. Which of the following sentences can be classified into statements?

d. In Unsyiah a student card is valid for 4 years and it can be extended.

38. From the following sentences, which one is not a statement?

c. If a, b, and c are the sides of the triangle ABC, then a² + b² = c².

39. Which one is a statement in the following sentences?

c.   If  y² = x, then y is not a function.

40. Which one is correct?

a.  The square of an even number is an even number and 3 + 2 £ 6.

41. Which statement is incorrect?

.

d.  A square is a rectangle and a kite is a square.

42. Which one is correct?

d.  A rectangle is a parallelogram and a square is a rhombus.

e.  A parallelogram is a square and a square is a rhombus.

43. Which statement is incorrect?

e.  A trapezoid is a rhombus or a kite.

44. The statement “If there is heavy rain today, I will not go to campus” is

       e. The converse of “If I will not go to campus, there will be heavy rain today”.

45. If we have to come at 07.00 in the campus, then we have to leave our home at 06.00. If we have to leave our home at 06.00, then we have to get up at 05.00. Therefore,

.

       c. If we have to come at 07.00 in the campus, then we have to get up at 05.00.

46. You know that I always stay in Bandung or Banda Aceh. You know also that Linda is always in Semarang or in  Pemalang. If you do not find me in Banda Aceh and you do not find her in Pemalang, what can you say?

   

       b. Yeah, I can say “You are in Bandung and she is in Semarang”.

47. Aida is in Berlin only if she is in holiday. She is in holiday only if she has finished her exam. We see that this month, she has finished her exam. What does it mean?

       a. It means that Aida is in holiday in Berlin.

48. If Ilham has finished his test, he should visit his uncle. If he visits his uncle, he has to give his uncle a fiction book. If he gives his uncle a fiction book, he has to buy the same book for himself. So, if Ilham does not have to buy a book which is the same as for his uncle, then …

       d. Ilham has not finished his test.

       e. Ilham does not visit his uncle and he has not finished his test.

49. The converse of “If you buy this orange, you will be given orange juice” is

       d. If you are not given orange juice, you will not buy this orange.

       e. If you buy this orange, you will not be given orange juice.

50. If General Susanto has a gun, he will use it only if he is robbed. The negation of this statement is

a.  Gen. Susanto has a gun, he is robbed, but he does not use the gun.

51. The negation of ”If the sun has lighted the earth, then the morning water droplets have made the soil wet” is

b.  The sun has lighted the earth, but the morning water droplets have not made the soil wet.

52. The statement “If the gate is closed, the train will cross over the gate immediately”  is logically equivalent to  

c.  If the train do not cross over immediately, the gate is not closed.

53. The conditional statement  a ® ~ b  is

e. The contrapositive of  b ® ~a.

54. The statement  ~ p  ®  q  is

      a. logically equivalent to  p Ú q

                       

55. Pursue me and I will catch you (Kejarlah daku, kau kutangkap). This statement indicates that

      d. If I do not catch you, then you do not pursue me.

      e. If you do not catch me, then I will catch you.

56. What is the other way of expression which is equivalent to “If you are honest, then I will be honest too” ?

.

      c. If I am not honest, you will not be honest.

      d. If you are not honest, I will not be honest.

      e. If I am not honest, you will be honest.

57. If someone says “I will not give you an umbrella or you give me a motor bike”. Based on implication, we can say that

      b. If I give you an umbrella, then you give me a motor bike.

58. If  soap is displayed (dipajang) near  tooth paste,  tooth paste is displayed near a toothbrush. If  tooth paste is displayed near a tooth brush, betadine antiseptic is displayed near betadine solution.  If betadine solution is displayed near betadine antiseptic, shampoo is displayed near body scrub.  In fact, shampoo is not displayed near body scrub. This indicates that,

a.  Tooth paste is not displayed near soap.

59. Hendi is a sailor and he always sails in Java sea. Rudi is a farmer and she always stays in a village. What can we conclude from these statements?

d.  Rudi is a farmer and Hendi is a sailor.

60. The statement “There is no day without happiness” is

b.  equivalent to the statement “All days are with happiness”.