Ngezindlela ezahlukene ukufakazela Theorem Pythagorean: Izibonelo, incazelo kanye nokubuyekeza

Enye into ngamakesita ayikhulu uqinisekile amaphesenti ukuthi umbuzo, okuyinto ilingana esigcawini hypotenuse, noma imuphi umuntu omdala ukuphendula ngesibindi: ". Isamba sezikwele imilenze" Lokhu theorem ziqine washaya izingqondo wonke umuntu abafundile, kodwa wena nje cela othile ukuba kufakazelwe, futhi kungase kube nezinkinga. Ngakho-ke, masikhumbule futhi ucabangele izindlela ezahlukene ukufakazela theorem kaPythagoras.

Ekuhlaziyeni biography

Theorem kaPythagoras kuba ezijwayele cishe wonke umuntu, kodwa ngasizathu simbe, ukuphila komuntu, okuyinto eye wenza ukuba ukukhanya, akuyona ithandwe. Lena fixable. Ngakho-ke, ngaphambi kokuba ukuhlola izindlela ezahlukene ukufakazela theorem kaPythagoras, kumelwe kafushane ebazi ubuntu bakhe.

Pythagoras - sefilosofi, isazi sezibalo, isazi sefilosofi odabuka eGrisi lasendulo. Namuhla kunzima kakhulu ukuhlukanisa yomlando wakhe kusukela ebezilokhu lisungulwe inkumbulo lo muntu omkhulu. Kodwa kusobala kwemisebenzi abalandeli bakhe, Pifagor Samossky wazalelwa esiqhingini Samos. Uyise wayengumGreki wokusika amatshe evamile, kodwa unina wayevela emkhayeni zasebukhosini.

Ngokwe-legend, ukuzalwa Pythagoras wabikezela owesifazane okuthiwa Pythia obubusa nodumo okuthiwa umfana. Ngokusho ukuqagela wakhe lokuzalwa yomntwana esingabulethela eziningi futhi izuzise zona ubuhle esintwini. Lokho eqinisweni akwenza.

Ukuzalwa theorem

Lapho esemusha, Pythagoras lisuswe kusuka Samos eGibhithe ukuhlangana zo waseGibhithe ezaziwayo. Ngemva kokuhlangana nabo, wabe esengeniswa kwalolu qeqesho, futhi wayazi ukuthi yonke impumelelo enkulu yalo mGibhithe ngefilosofi, izibalo kanye imithi.

Kwaba cishe eGibhithe Pythagoras liphefumulelwe nobukhosi nobuhle kwemibhoshongo yaseGibhithe futhi wadala bayihlaba kakhulu inkolelo yakhe enkulu. Kungase ukwethusa abafundi, kodwa izazi-mlando zanamuhla bakholelwa ukuthi Pythagoras ayizange bayihlaba kakhulu inkolelo yakhe. Futhi kuphela wanginika ulwazi lakhe labalandeli okwathi kamuva igcwaliswe zonke izibalo ezidingekayo zezibalo.

Kungakhathaliseki ukuthi laliyini, manje eyaziwa ezingaphezu kwesisodwa indlela balokhu theorem, kodwa abaningana. Namuhla Akekho owaziyo ukuthi amaGreki Wenza izibalo zabo, ngakho kunezindlela ezihlukahlukene ukubheka ubufakazi theorem kaPythagoras.

Pythagoras 'theorem

Ngaphambi kokuqala noma imuphi umsebenzi ukubala, udinga ukuthola theory ukufakazela. Theorem kaPythagoras yilesi: "Ngo-unxantathu lapho omunye engeli ^imayelana 90, isamba sezikwele imilenze lilingana esigcawini hypotenuse."

Sekukonke kukhona 15 ngezindlela ezahlukene ukufakazela theorem kaPythagoras ukuphi. Lena sibalo kunalokho okusezingeni eliphezulu, kanjalo unake ethandwa kakhulu babo.

indlela eyodwa

Okokuqala, iveze ukuthi sinikwa. Lezi idatha uzobe enwetshiwe kwezinye izindlela ubufakazi theorem kaPythagoras, ngakho kuyafaneleka ukukhumbula zonke namagama akhona.

Lokucabanga unikezwa unxantathu wesokudla-angled nge imilenze, futhi hypotenuse elilingana c. Indlela yokuqala lusekelwe ebufakazini ukuthi, ngenxa unxantathu wesokudla ezidingekayo ukuqedela isikwele.

Ukuze wenze lokhu, udinga ubude umlenze ingxenye alinganayo ukuqedela umlenze, futhi okuphambene nalokho. Ngakho-ke kufanele sibe izinhlangothi ezimbili alinganayo isikwele. Singagcina ngokubu- ukudweba imigqa emibili okufanayo, futhi isikwele usulungile.

Ngaphakathi, izibalo okuholela Kumele udwebe esinye square nohlangothi ilingane hypotenuse unxantathu yasekuqaleni. Kulokhu vertices of ac nokuxhumana kuyadingeka ukudweba izingxenye ezimbili elilinganayo ngokulinganisa. ukuthola Ngakho ezinhlangothini ezintathu isikwele, omunye okuyinto unxande yasekuqaleni bocalantsatfu le hypotenuse. Docherty uhlala kuphela ingxenye yesine.

Ngokusekelwe iphethini okuholela ke kungenziwa baphetha ngokuthi endaweni yangaphandle kwesikwele ilingana (a + b) ^2. Uma ubheka izibalo, ungabona ukuthi ngaphezu isikwele kwangaphakathi it has onxantathu ezine kwesokudla angled. Endaweni ngayinye 0,5av.

Ngakho-ke, endaweni ilingana: 4 * 0,5av + c ² = ² + 2av

Ngakho, (b +) ² = c ² + 2av

Futhi-ke, nge ² = ² + ²

Lokhu kufakazela theorem.

Indlela ezimbili: onxantathu efanayo

Lokhu Ifomula ubufakazi theorem kaPythagoras elisuselwa ngesisekelo imvume geometry ingxenye lezi onxantathu. Lithi imilenze unxantathu wesokudla - the ezilinganiselwe isilinganiso hypotenuse yayo futhi ubude hypotenuse, lovela vertex ^90.

Idatha kokuqala ziyafana, ngakho ake uqale ngokushesha ne ubufakazi. Dweba perpendicular ohlangothini ingxenye AB CD. Ngokusekelwe ukwamukelwa ngenhla imilenze onxantathu bayalingana:

AC = √AV * AD, CB = √AV * DV.

Ukuze siphendule umbuzo kanjani ukufakazela theorem kaPythagoras, ubufakazi kufanele sidluliselwe nge squaring kokubili nokungalingani.

AC ² = AB * BP kanye CB ² = AB * DV

Manje udinga ukwengeza i-ukungalingani okuholela.

AU ^{2 2} + CB = AB * (BP * ET) lapho BP = AB + ET

It kuvela ukuthi:

AC ² + ² = CB AB * AB

Futhi-ke:

AU ^{2 2} + CB = AB ²

Ubufakazi theorem kaPythagoras nezindlela ezahlukene isixazululo yayo kudingeka ukuba yehlukene kaningi okubalwa indlela kule nkinga. Nokho, le ndlela ingenye elula.

Enye indlela yokubala

Incazelo ngezindlela ezahlukene ukufakazela Pythagorean Theorem kungenzeka wathula, inqobo nje uma iningi musa ngokwabo baye baqala ukuqhuba. Abaningi amathekhiniki akuhileli nje izibalo, kodwa futhi ukwakhiwa unxantathu yasekuqaleni izibalo entsha.

Kulokhu-ke kuyadingeka ukuze uqedele umlenze BC lomunye unxantathu wesokudla-angled le IRR. Ngakho manje asekhona onxantathu ezimbili umlenze Welanga ezivamile

Ukwazi ukuthi izindawo izibalo efanayo ube isilinganiso njengoba izigcawu Ubukhulu yabo efanayo komugqa ke:

S _ABC * ² - S ² * _HPA = S * futhi _AVD ² - S ² * a _VSD

_Abc * S ⁽² -c ²⁾ = ² * (S _AVD -S _VVD)

Lokudlulisa ^{2 2} = ²

² = ² + ²

Ngenxa yezindlela ezahlukene ubufakazi theorem kaPythagoras ebangeni 8, le ndlela kufanelekile neze, ungasebenzisa inqubo elandelayo.

Indlela elula ukufakazela theorem kaPythagoras. Izibuyekezo

Kunenkolelo-mlando, le ndlela yaqale osetshenziselwa ubufakazi theorem eGrisi lasendulo. He is elula njengoba akudingi nhlobo yokukhokha. Uma usondela isithombe kahle, ubufakazi egomela ukuthi ² + ² = c ^2, uzobe ke bubonakala ngokucacile.

Imigomo nemibandela ye-le nqubo kuyoba ukuhluka lowo odlule. Ukuze afakazele theorem, ucabange ukuthi unxantathu kwesokudla angled ABC - isosceles.

Hypotenuse AC zilawule isiqondiso kwesikwele futhi docherchivaem ezinhlangothini zalo ezintathu. Ngaphandle kubalulekile ukuchitha imigqa emibili idayagonali ukwakha isikwele. Ngakho, ukuze uthole onxantathu ezine equilateral ngaphakathi kwalo.

Ngu Catete AB futhi CD njengoba kudingeka Docherty on esigcawini bese ubamba on line elilodwa idayagonali ngamunye kubo. Dvweba umugca kusuka ngowokuqala vertex A, owesibili - kusuka C.

Manje kudingeka sisithathe kubukisisa isithombe. Njengoba hypotenuse AC onxantathu ezine ilingane yokuqala, kodwa Catete ezimbili, likhuluma ngokuphela ukunemba yalesi theorem.

By the way, sibonga le ndlela, ubufakazi theorem kaPythagoras, futhi wazalwa ibinzana adumile: ". Ibhulukwe Pythagorean kuzo zonke izinhlangothi bayalingana"

J. Ubufakazi. Garfield

Dzheyms Garfild - uMongameli ngolwamashumi amabili e-United States of America. Ngaphezu kwalokho, uye washiya uphawu yakhe emlandweni njengombusi waleli-United States, wayebuye abe self-wafundisa onekhono.

Ekuqaleni kwenkonzo yakhe, bekanguthishela njalo esikoleni abantu, kodwa ngokushesha ngaba umqondisi omunye zemfundo ephakeme. Isifiso self-ukuthuthukiswa kanye kwamenza ukuphakamisa umbono omusha we ubufakazi theorem ka Pythagoras. Theorem futhi isibonelo isixazululo yayo kanje.

Okokuqala kubalulekile ekudwebeni unxantathu iphepha ezimbili wamaqhuqhuva umlenze ukuze omunye okwakukhona phambili yokugcina. Vertices yalezi onxantathu kufanele exhunywe bagcina ukuthola esishwibweni.

Njengoba yaziwa, kwendawo trapezoid ilingana umkhiqizo nenxenye isamba kwesisekelo salo futhi ukuphakama.

S = a + b / 2 * (a + b)

Uma sicabangela trapezoid okuholela, njengoba isibalo sakhiwa onxantathu ezintathu, endaweni yalo ingatholakala kanje:

S = aw / 2 * 2 + ^2/2

Manje kubalulekile nokulinganisa inkulumo ethi ezimbili zokuqala

2av / 2 + c / 2 = (a + b) ^2/2

² = ² + ²

Mayelana Pythagoras nendlela ukufakazela awukwazi ukubhala eyodwa ivolumu ebhukwinitifundvo. Kodwa ucabanga ukuthi kunengqondo uma lolo lwazi angeke ekusebenzeni?

ngokoqobo theorem kaPythagoras

Ngeshwa, esikhathini esikoleni esivamile yesimanje sihlinzekela ukusetshenziswa kwalesi theorem kuphela izinkinga weJiyomethri. Abaphothule kulesi sikole ngokushesha ukushiya izindonga esikoleni, futhi bengazi, begodu bonyana bangenza bunjani basebentisa lwati lwabo kanye namakhono practice.

Empeleni, ukusebenzisa theorem kaPythagoras ekuphileni kwabo kwansuku zonke can ngamunye. Futhi hhayi kuphela umsebenzi okhokhelwayo, kodwa futhi abavamile yasendlini. Cabangela izibonelo ezimbalwa lapho theorem kaPythagoras nendlela ukufakazela kungaba kudingekile kakhulu.

theorems Ukuxhumana kanye astronomy

Kunengqondo ukuthi kungahlanganiswa izinkanyezi onxantathu ephepheni. Empeleni, ukufunda izinkanyezi - endaweni wezesayensi lapho kabanzi theorem kaPythagoras.

Ngokwesibonelo, cabanga ukunyakaza ugongolo ukukhanya emkhathini. Kuyaziwa ukuthi ukukhanya luhamba zombili izinkomba ngezinga elifanayo. AB trajectory, okuyinto ushukumisela wokukhanya ibizwa ngokuthi l. Futhi ingxenye yesikhathi adingekayo ukukhanya ukusuka endaweni A ukhombe B, sikubiza t. Futhi ijubane ugongolo - c. It kuvela ukuthi: c * t = l

Uma ubheka le ugongolo efanayo enye indiza, isibonelo, umkhumbi isikhala, esishukumisela nge v isivinini ke ngaphansi lezo zidumbu Uyobe uqondiswa guqula isivinini sabo. Nokho, ngisho izakhi fixed izoya nge v Isivinini kolunye uhlangothi.

Ake sithi liner yamahlaya elintantayo kwesokudla. Khona-ke amaphuzu A no-B, okuyinto idabukile phakathi ugongolo kuzoya ngakwesokunxele. Ngaphezu kwalokho, lapho uhamba ugongolo ukusuka endaweni A ukhombe B, ukhombe isikhathi ukuhambisa, futhi, ngokufanelekile, ukukhanya kufikile iphuzu C. entsha Ukuthola isigamu ibanga lapho endaweni A uye wathuthela, kubalulekile ukuba ngandise ijubane umkhumbi ngesigamu isikhathi ugongolo travel (t ').

d = t '* v

Futhi ukuthola kwaba kude kangakanani ngaleso sikhathi bakwazi ukudlula umsebe wokukhanya kuyadingeka ukumaka iphuzu maphakathi we-beech entsha s nenkulumo ezilandelayo:

s = c * t '

Uma sicabange ukuthi iphuzu ukukhanya C no-B, kanye umkhumbi isikhala - phezulu unxantathu isosceles, ingxenye kusukela iphuzu A kuya liner ngeke uluhlukanise phakathi onxantathu ezimbili kwesokudla-angled. Ngakho-ke, sibonga theorem kaPythagoras ungathola ibanga ukuthi wakwazi ukudlula umsebe wokukhanya.

s = l ^{2 2} + d ²

Lokhu isibonelo, yebo, hhayi kakhulu kubancono, ngobe ambalwa nje zingaba lucky ngokwanele ukuyizama practice. Ngakho-ke, sicabanga zokusebenza ezivamile ngaphezulu kwalesi theorem.

Ububanzi mobile isignali ukudluliswa

kusukela epilweni yangamalanga yesimodeni akunakwenzeka ukucabanga ngaphandle ukhona smartphone. Kodwa bangaki kubo kuzofuneka ukuthi PROC uma asikwazi ukuxhuma ababhalisile ngokusebenzisa mobile?!

mobile izinga ukuxhumana kuncike ngqo ukuphakama lapho antenna ukuba opharetha mobile. Ukuze sibone ukuthi kude imibhoshongo iselula bangathola isiginali, ungasebenzisa theorem kaPythagoras.

Ake sithi ufuna ukuthola ukuphakama okucishe umbhoshongo fixed, ukuze ukusabalalisa isignali endaweni engaba amakhilomitha angu-200.

AB (ukuphakama umbhoshongo) = x;

Sun abangakudla (isikhathi sendlala engaba) = km 200;

OC (engaba emhlabeni) = 6380 km;

lapha

OB = OA + AVOV = r + x

Ukusebenzisa theorem kaPythagoras, siyathola ukuthi ubuncane umbhoshongo ukuphakama kufanele kube 2.3 amakhilomitha.

ifayela echaza ifomu Pythagorean ekhaya

Isimanga siwukuthi theorem kaPythagoras kungaba wusizo ngisho nasezintweni zasekhaya ezifana ukuzimisela ukuphakama egumbini lesigungu isibonelo. Uma uthi nhlá, asikho isidingo ukusebenzisa izibalo eziyinkimbinkimbi enjalo, ngoba ungavele thatha izilinganiso zakho nge ithephu. Kodwa abaningi bayazibuza ukuthi kungani inqubo Yakha kukhona izinkinga ezithile, uma zonke izilinganiso athathwa phezu ncamashi.

Iqiniso liwukuthi ekamelweni kwenzekani endaweni evundlile base beqala ikhwele odongeni. Ngakho-ke, odongeni ohlangothini iKhabhinethi in inqubo oqeda design kumele flow ngokukhululekile futhi ukuphakama, futhi izikhala idayagonali.

Ake sithi ikhabethe 800 mm ukujula. Ibanga kusuka phansi kuya ophahleni - 2600 mm. Abanolwazi iKhabhinethi umenzi uthi ukuphakama ebiyelwe kufanele bube 126 mm esingaphansi ukuphakama egumbini. Kodwa kungani ku 126mm? Cabangela isibonelo esilandelayo.

Ngaphansi Ubukhulu ekahle iKhabhinethi uzohlola isenzo Theorem Pythagorean:

√AV AC = ² + ² √VS

AU = √2474 ² 800 ² = 2600 mm - bonke bukhomba.

Ake sithi, ukuphakama iKhabhinethi akulingani 2474 mm 2505 mm. Khona-ke:

AU = √2505 ² + √800 = 2629 mm ^2.

Ngenxa yalokho, lo iKhabhinethi asifanele ukufakwa egumbini. Kusukela lapho wacosha isikhundla salo uqotho kungabangela ukulimala emzimbeni wakhe.

Mhlawumbe kubhekwe izindlela ezahlukahlukene ukufakazela Pythagorean Theorem ososayensi ezahlukene, singaphetha ngokuthi ingaphezu kweqiniso. Manje ungasebenzisa ulwazi ekuphileni kwabo kwansuku zonke, futhi ngokuphelele ukuthi zonke izibalo akuzona kuphela ewusizo, kodwa futhi kuyiqiniso.