Wp/iba/Teorem Pythagoras

Teorem Pythagoras
Type	Teorem
Field	Geometri Euclid
Statement	The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c).
Symbolic statement
Generalizations	Law of cosines; Solid geometry; Non-Euclidean geometry; Differential geometry;
Consequences	Pythagorean triple; Reciprocal Pythagorean theorem; Complex number; Euclidean distance; Pythagorean trigonometric identity;

Dalam matematik, teorem Pythagoras iya nya kaul paung dalam geometri Euclid entara tiga piak segitiga besukut bujur. Iya madahka pemesai kuadrat ti sepiak iya nyadi hipotenus (sepiak ti betentang enggau sukut bujur tauka 90°) sama enggau penyampau pemesai kuadrat ba dua iti tisi bukai.

Teorem tu tau ditulis nyadi persamaan ti bekaul enggau pemanjai tisi a, b enggau hipotenus c, kekadang dikumbai penyama Pythagoras.:^[1]

a^{2}+b^{2}=c^{2}.

Teorem tu diberi nama ari siku raban filosofi Gerika iya nya Pythagoras, ke ada urung taun 570 SM. Teorema tu udah dibuktika mayuh kali ngena mayuh macham chara – nyangka pemadu mayuh ungkup sebarang teorem matematik. Bukti nya mayuh macham, nyengkaum bukti geometri enggau bukti algebra, lalu sekeda nya udah beribu taun ke udah.

Lebuh ruai Euclid diarika sistem koordinat Cartes dalam geometri analitik, penyauh Euclidean ngemuaska kaul Pythagoras: penyauh kuadrat entara dua titik sama enggau penyampau kuadrat bida dalam tiap-tiap koordinat entara titik.

Teorem tu ulih dipeda enggau mayuh macham chara: ngagai ruang ti tinggi agi dimensyen, ngagai ruai ti ukai Euclid, ngagai objek ti ukai segitiga besukut bujur, enggau ngagai utai ti ukai segitiga sekali tang pepejal n-dimensyen.

Bukti ngena segi empat ti digaga

Rearrangement proof of the Pythagorean theorem.
(The area of the white space remains constant throughout the translation rearrangement of the triangles. At all moments in time, the area is always c². And likewise, at all moments in time, the area is always a²+b².)

Bukti tusun baru

Dalam siti bukti nusun baru, dua iti kuadrat dikena ti tisi iya bisi ukur $a+b$ enggau ti ngundan empat segitiga lurus ti tisi iya dilabel enggau a, b sereta c, lalu c nya hipotenus. Ba segi empat ba sepiak kanan, segitiga diengkah ngambika sukut segi empat nya setipak enggau sukut sukut kanan ba segitiga, lalu nempa segi empat ba tengah ti tisi iya sepemanjai c. Tiap iti pitak luar bisi pemesai $(a+b)^{2}$ seraya $2ab+c^{2}$ , enggau $2ab$ ngayanka semua pemesai empat segitiga nya. Dalam segi empat ti besai ba sepiak kiba, empat segitiga nya dipindahka ngambika nyadi dua segi empat ti sebaka enggau tisi pemanjai a enggau b. Segi empat tebuka tu ba penuduk baru sida diatu udah ngaga garis dua iti segi empat baru, siti ngembuan pemanjai tisi a ditempa ba sukut baruh-kiba, lalu siti agi segi empat pemanjai tisi b ditempa ba sukut atas-kanan. Dalam penuduk baru tu, sepiak kiba tu diatu bisi kuasa dua ari pemesai $(a+b)^{2}$ sereta $2ab+a^{2}+b^{2}$ . Ketegal kedua-dua pitak nya ngembuan pemesai $(a+b)^{2}$ iya nitihka ukur bukai ari luas persegi mega sama enggau pangan diri ngambika $2ab+c^{2}$ = $2ab+a^{2}+b^{2}$ . Enggau pemesai empat segitiga ti dibuai ari kededua piak equation utai ti agi tinggal iya nya $a^{2}+b^{2}=c^{2}.$ ^[2]

Dalam siti agi bukti segi empat tebuka dalam kutak kedua mega tau diengkah ngambika kededua bisi siti sukut ti setipak enggau sukut ti bejurut ba segi empat. Enggau chara tu sida mega nempa dua iti kutak, sekali tu ba sukut ti bejurut, enggau pemesai $a^{2}$ enggau $b^{2}$ ti deka sekali agi ngujungka segi empat kedua enggau pemesai $2ab+a^{2}+b^{2}$ .

Pakar matematik menua Inggeris Sir Thomas Heath meri bukti tu dalam komen iya pasal Proposal I.47 dalam Elemen Euclid, lalu nyebut perambu ari ahli matematik Jerman Carl Anton Bretschneider enggau Hermann Hankel ti madahka Pythagoras engka nemu bukti tu. Heath empu nyukung perambu ti bebida ke bukti Pythagoras, tang ngaku ari pun randau iya "litaricha Yunani ti dikembuan kitai enggi lima abad keterubah pengudah Pythagoras nadai ngundan penerang ti netapka pekara tu tauka sebarang penemu geometri ti besai ngagai iya."^[3] Pansik ke baru-baru tu udah ngasuh orang majak ragu-ragu ba sebarang pengawa Pythagoras nyadi penengkebang matematik, taja pan debat pasal pekara tu majak nyadi.^[4]

Bukti algebra

Diagram of the two algebraic proofs

Teorema tu ulih dibuktika ngena chara algebra ngena empat salin segitiga ti sama ti ditusun enggau simetrik ngelingi segi empat ti bisi tisi c, baka ti dipandangka ba bagi baruh diagram.^[5] Tu ngujungka segi empat ti besai agi, enggau tisi a + b enggau pemesai (a + b)². Empat segitiga enggau tisi segi empat c patut ngembuan pemesai ti sama enggau segi empat ti besai agi,

(b+a)^{2}=c^{2}+4{\frac {ab}{2}}=c^{2}+2ab,

meri asil

c^{2}=(b+a)^{2}-2ab=b^{2}+2ab+a^{2}-2ab=a^{2}+b^{2}.

Bukti ti sebaka ngena empat salin segitiga kanan ti bisi tisi a, b enggau c, ditusun dalam segi empat ti bisi tisi c baka ba setengah atas diagram.^[6] Segitiga nya sebaka enggau pemesai ${\tfrac {1}{2}}ab$ , lalu segi empat ke mit ngembuan pemanjai tisi b − a enggau pemesai (b − a)². Nya alai, pemesai segi empat ke besai nyadi:

(b-a)^{2}+4{\frac {ab}{2}}=(b-a)^{2}+2ab=b^{2}-2ab+a^{2}+2ab=a^{2}+b^{2}.

tang tu segi empat ke ngembuan pemanjai tisi c enggau pemesai c², nyadi

c^{2}=a^{2}+b^{2}.

Penerang

↑ Judith D. Sally; Paul Sally (2007). "Chapter 3: Pythagorean triples". Roots to research: a vertical development of mathematical problems. American Mathematical Society Bookstore. p. 63. ISBN 978-0-8218-4403-8.
↑ Benson, Donald. The Moment of Proof : Mathematical Epiphanies, pp. 172–173 (Oxford University Press, 1999).
↑ Template:Wp/iba/Harvtxt, pp. 351–352
↑ Huffman, Carl (23 February 2005). "Pythagoras". In Zalta, Edward N. (ed.). The Stanford Encyclopedia of Philosophy (Winter 2018 Edition)., "It should now be clear that decisions about sources are crucial in addressing the question of whether Pythagoras was a mathematician and scientist. The view of Pythagoras's cosmos sketched in the first five paragraphs of this section, according to which he was neither a mathematician nor a scientist, remains the consensus."
↑ Alexander Bogomolny. "Cut-the-knot.org: Pythagorean theorem and its many proofs, Proof #4". Cut the Knot. Diambi 4 November 2010.
↑ Alexander Bogomolny. "Cut-the-knot.org: Pythagorean theorem and its many proofs, Proof #3". Cut the Knot. Diambi 4 November 2010.

[Sally0-1] Judith D. Sally; Paul Sally (2007). "Chapter 3: Pythagorean triples". Roots to research: a vertical development of mathematical problems. American Mathematical Society Bookstore. p. 63. ISBN 978-0-8218-4403-8.

[2] Benson, Donald. The Moment of Proof : Mathematical Epiphanies, pp. 172–173 (Oxford University Press, 1999).

[3] Template:Wp/iba/Harvtxt, pp. 351–352

[4] Huffman, Carl (23 February 2005). "Pythagoras". In Zalta, Edward N. (ed.). The Stanford Encyclopedia of Philosophy (Winter 2018 Edition)., "It should now be clear that decisions about sources are crucial in addressing the question of whether Pythagoras was a mathematician and scientist. The view of Pythagoras's cosmos sketched in the first five paragraphs of this section, according to which he was neither a mathematician nor a scientist, remains the consensus."

[5] Alexander Bogomolny. "Cut-the-knot.org: Pythagorean theorem and its many proofs, Proof #4". Cut the Knot. Diambi 4 November 2010.

[rotate-6] Alexander Bogomolny. "Cut-the-knot.org: Pythagorean theorem and its many proofs, Proof #3". Cut the Knot. Diambi 4 November 2010.

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