Uwerenge kumathandiza kupeza ngodya zabwino za makona a. The Lingaliro Lovomerezeka pa Uwerenge kumathandiza kupeza ngodya zabwino za makona ndi

The makona ndi polygon ndi mbali zitatu (kumathandiza kupeza ngodya zitatu). Nthawi zambiri, gawo amaimira zilembo zazing'ono lolingana zilembo zazikulu, akuimira mfundo zosiyana. M'nkhaniyi tiona tione mitundu mosiyanasiyana zojambula, Lingaliro Lovomerezeka, lomwe liri kutanthauzira ndi wofanana Uwerenge kumathandiza kupeza ngodya zabwino za makona a.

Mitundu kumathandiza kupeza ngodya zabwino yaikulu

Zotsatirazi mitundu ya polygon ndi mfundo zitatu:

• pachimake-angled, imene kumathandiza kupeza ngodya zabwino zonse lakuthwa;
• amakona anayi ndi imodzi mlingo woyenerera, mbali analipanga, anatchula miyendo, ndi mbali imene oyenerera wotsutsa ku mlingo woyenerera amatchedwa hypotenuse;
• obtuse pamene wina ngodya ndi obtuse ;
• isosceles, amene mbali ziwiri ndi ofanana ndipo iwo amatchedwa lateral, ndi wachitatu - kansalu ndi tsinde ndi;
• equilateral ndi zitatu mbali ofanana.

katundu

Kuonetsetsa katundu zikuluzikulu zomwe ndi khalidwe la mtundu uliwonse wa makona:

• popenyana ndi mbali wamkulu nthawi zonse wamkulu njingayo, ndi mosemphanitsa;
• ali kumathandiza kupeza ngodya zabwino ofanana popenyana ndi ofanana-waukulu chipani, ndi mosemphanitsa;
• mu makona aliyense ali kumathandiza kupeza ngodya ziwiri pachimake;
• njingayo akunja kuposa aliyense ngodya mkati osati moyandikana thereto;
• Uwerenge kumathandiza kupeza ngodya ziwiri nthawi zonse madigiri zosakwana 180;
• kunja ngodya liri Uwerenge ngodya ena awiri, omwe si mezhuyut naye.

The Lingaliro Lovomerezeka pa Uwerenge kumathandiza kupeza ngodya zabwino za makona ndi

The Lingaliro Lovomerezeka limanena kuti ngati inu kuwonjezera ngodya zonse za mawonekedwe zojambula, limene lili mu ndege Euclidean, ndiye ndalama zawo zidzakhala madigiri 180. Tiyeni ayeserere Lingaliro Lovomerezeka izi.

Tiyeni ife ndi makona umasinthasintha ndi mfundo KMN. Kudutsa pamwamba pa M adzakangamira zikufanana mwachindunji kwa mzere KN (ngakhale mzere uwu umatchedwa Euclid). Tisaiwale mfundo A kotero kuti mfundo K ndi imakonzedwa ku mbali zosiyana za mzere MN. Ife tifika mbali womwewo wa AMS ndi MUF, amene, monga mkati, kunama crosswise kupanga intersecting MN molumikizana ndi CN mwachindunji ndi MA, amene yomweyi. Kuchokera pamenepa, ndizoonekeratu kuti Uwerenge kumathandiza kupeza ngodya zabwino za makona ndi, inali pa mfundo ya M ndipo N ndi wofanana kukula kwa CMA njingayo. kumathandiza kupeza ngodya zabwino atatu zigwirizana wa ndalama wofanana Uwerenge kumathandiza kupeza ngodya zabwino za KMA ndi MCS. Popeza deta ndi kumathandiza kupeza ngodya zabwino mkati wachibale amaganiza kufanana mizere CL ndi masentimita MA pa intersecting, ndalama zawo ndi madigiri 180. Izi zikutsimikizira pa Lingaliro Lovomerezeka.

chifukwa

Pamwambapa Lingaliro Lovomerezeka pamwambapa akusonyeza corollary zotsatirazi: iliyonse makona ali kumathandiza kupeza ngodya ziwiri pachimake. Pofuna kutsimikizira zimenezi, tiyeni tinene kuti chithunzi ichi geometrical ali chimodzi chokha pachimake oyenerera. Mukhozanso kuganiza kuti palibe ngodya si lakuthwa. Pankhaniyi ayenera kukhala kumathandiza kupeza ngodya zabwino osachepera awiri, kukula kwa amene ali ofanana ndi wamkulu kuposa madigiri 90. Koma ndiye Uwerenge kumathandiza kupeza ngodya zabwino ndi wamkulu kuposa madigiri 180. Koma izi sizingatheke, monga malinga ndi Lingaliro Lovomerezeka ndalama kumathandiza kupeza ngodya zabwino za makona ndi wofanana 180 ° - kenanso, chimodzimodzi. Ndicho chimene ankafunika kupereka umboni.

Katundu kunja ngodya

ndi Uwerenge kumathandiza kupeza ngodya zabwino za makona, amene ali kunja chiyani? Yankho la funso limeneli akamagwira ntchito imodzi mwa njira ziwiri. Woyamba ndi kuti muyenera kupeza Uwerenge kumathandiza kupeza ngodya zabwino, zimene imodzi iliyonse vertex, ndiko kuti, angelo atatu. The chachiwiri zikutanthauza kuti muyenera kupeza Uwerenge kumathandiza kupeza ngodya zabwino asanu pa mfundo ya. Kuthana ndi chiyambi cha chimake choyamba. Choncho kansalu lili asanu ngodya akunja - pamwamba pa ziwiri. Mugulu ali kumathandiza kupeza ngodya zabwino ofanana pakati pawo, popeza ofukula:

∟1 = ∟4, ∟2 = ∟5, ∟3 = ∟6.

Komanso, amadziwika kuti ngodya lakunja la makona ndi wofanana Uwerenge awiri mkati, zimene siziri mezhuyutsya naye. Choncho,

∟1 = ∟A + ∟S, ∟2 = ∟A + ∟V, ∟3 = ∟V + ∟S.

Kuchokera pamenepa, zikuoneka kuti Uwerenge kumathandiza kupeza ngodya zabwino kunja, omwe atengedwa mmodzimmodzi pafupi wina vertex adzakhala wofanana:

∟1 + ∟2 + ∟3 = ∟A + + ∟S ∟A ∟V + + + ∟V ∟S = 2 × (∟A + ∟V ∟S +).

Popeza chakuti Uwerenge kumathandiza kupeza ngodya zabwino ndi wofanana madigiri 180, akhoza kutsutsa kuti ∟A + ∟V ∟S = + 180 °. Izi zikutanthauza kuti ∟1 + ∟2 + ∟3 = 2 × 180 ° = 360 °. Ngati Yachiwiri ntchito, Uwerenge kumathandiza kupeza ngodya zabwino asanu adzakhala safanana wamkulu kawiri. Ie Uwerenge kumathandiza kupeza ngodya zabwino za makona kunja adzakhala:

∟1 + ∟2 + ∟3 + ∟4 + ∟5 + ∟6 = 2 × (∟1 + ∟2 + ∟2) = 720 °.

makona bwino

Kodi ndi wofanana Uwerenge kumathandiza kupeza ngodya zabwino za kansalu lamanja, ndi chilumba? Yankho ndi, kachiwiri, kuchokera Lingaliro Lovomerezeka, limanena kuti kumathandiza kupeza ngodya zabwino za makona ndi kuwonjezera kwa madigiri 180. A phokoso zonena zathu (chuma) motere: mu kansalu bwino kumathandiza kupeza ngodya zabwino lakuthwa kuwonjezera kwa madigiri 90. Timasonyeza kulondola kwake. Pakhale anapatsidwa makona KMN, amene ∟N = 90 °. Ndi kofunika kutsimikizira kuti ∟K ∟M = + 90 °.

Choncho, malinga ndi Lingaliro Lovomerezeka pa Uwerenge kumathandiza kupeza ngodya zabwino ∟K + ∟M ∟N + = 180 °. Mu chikhalidwe ichi zikunenedwa kuti ∟N = 90 °. Likukhalira ∟K ∟M + + 90 ° = 180 °. Kuti ndi ∟K ∟M + = 180 ° - 90 ° = 90 °. Ndicho chimene ife tiyenera kutsimikizira.

Kuwonjezera katundu pamwamba pa kansalu zoyenera, mungathe kuwonjezera izi:

• kumathandiza kupeza ngodya zabwino, omwe amanama ndi miyendo ndi lakuthwa;
• ndi hypotenuse wa triangular woposa aliyense wa miyendo;
• Uwerenge miyendo kuposa hypotenuse ya;
• mwendo wa makona, limene lili moyang'anizana ndi mbali ya madigiri 30, theka la hypotenuse, kuti ndi wofanana ndi theka lake.

Monga katundu wina wa mawonekedwe zojambula akhoza kuwasiyanitsa Pythagorean Lingaliro Lovomerezeka. Iye ananena kuti kansalu ndi makona a madigiri 90 (amakona anayi), Uwerenge mabwalo miyendo liri lalikulu la hypotenuse lapansi.

Uwerenge kumathandiza kupeza ngodya ya makona isosceles

Poyamba paja ananena kuti makona isosceles ndi polygon ndi mfundo zitatu, munali awiri mbali ofanana. katundu Ichi chikudziwika geometrical chithunzi: ndi kumathandiza kupeza ngodya zabwino pa maziko ake ofanana. Tiyeni titsimikizire izi.

Tengani makona KMN, amene ali isosceles, SC - maziko ake. Timafunika kutsimikizira kuti ∟K = ∟N. Choncho, tiyeni tinene kuti MA - KMN ndi bisector wa makona wathu. ICA makona ndi chizindikiro cha kufanana ndi makona MNA. Ndicho, ndi kungoganizira anapatsidwa kuti masentimita = NM, MA ndi mbali wamba, ∟1 = ∟2, chifukwa MA - bisector izi. Ntchito kufanana kwa Triangles awiri, amene anganyoze kuti ∟K = ∟N. Choncho, Lingaliro Lovomerezeka imatsimikizirika.

Koma ndife chidwi, kodi Uwerenge kumathandiza kupeza ngodya zabwino za kansalu (isosceles). Chifukwa chikutanthauza alibe maonekedwe ake, tidzakhala kuyamba ku Lingaliro Lovomerezeka takambirana kale. Ndiko kuti, tikhoza kunena kuti ∟K + ∟M ∟N + = 180 °, kapena 2 × ∟K ∟M + = 180 ° (monga ∟K = ∟N). Izi sadzakhala katundu, monga Lingaliro Lovomerezeka pa Uwerenge kumathandiza kupeza ngodya zabwino za makona anali atatsimikizira kale.

Kupatula katundu wa nkhani za m'makona a makona ndi palinso mawu ofunika:

• mu ndi equilateral makona msinkhu, amene anali kufika m'munsi, ndi imodzi ya bisector apakatikati pa ngodya umene uli pakati pa mbali ofanana ndi olamulira chosokonekera maziko ake;
• apakatikati (bisector, okwera) imene ikuchitikira ku mbali ya chithunzi zojambula, ofanana.

equilateral makona

Limatchedwanso lamanja, ndi makona atatu, amene ali wofanana ndi magulu onse. Choncho ofanana ndi kumathandiza kupeza ngodya zabwino. Aliyense wa iwo ndi madigiri 60. Tiyeni tisonyeze chuma ichi.

Tiyeni tinene kuti tili kansalu KMN. Tikudziwa kuti Km = HM = KH. Izi zikutanthauza kuti, malinga ndi chuma cha kumathandiza kupeza ngodya zabwino ndi ili pa maziko a mu equilateral makona ∟K = ∟M = ∟N. Popeza, malinga Uwerenge kumathandiza kupeza ngodya zabwino za kansalu Lingaliro Lovomerezeka ∟K + ∟M ∟N + = 180 °, ndiye × 3 = 180 ° ∟K kapena ∟K = 60 °, ∟M = 60 °, ∟N = 60 °. Choncho, mfundo imatsimikizirika. Malinga ndi umboni pamwamba zochokera Lingaliro Lovomerezeka pamwamba, Uwerenge kumathandiza kupeza ngodya ya makona equilateral, monga Uwerenge kumathandiza kupeza ngodya zabwino za makona ina ndi madigiri 180. Kachiwiri kutsimikizira Lingaliro Lovomerezeka izi si koyenera.

Pali ena katundu khalidwe la makona equilateral:

• apakatikati bisector msinkhu chithunzi geometrical zofanana, ndipo kutalika kwake kuchita masamu monga (a × √3): 2;
• ngati polygon izi circumscribing bwalo, ndiye utali wozungulira adzakhala wofanana (a × √3): 3;
• ngati alembedwa bwalo equilateral makona, utali wozungulira yace (a × √3): 6;
• Malo chithunzi zojambula kuchita masamu mfundo: (A2 × √3): 4.

Obtuse makona

Ndi tanthauzo, ndi obtuse-angled makona, mmodzi wa m'makona ake ali pakati pa madigiri 90 mpaka 180. Koma anapatsidwa chakuti ena kumathandiza kupeza ngodya ziwiri za mawonekedwe zojambula lakuthwa, Ndizachidziwikire kuti iwo musalumphe madigiri 90. Choncho, Uwerenge kumathandiza kupeza ngodya zabwino za kansalu Lingaliro Lovomerezeka ntchito mu kuwerengetsa Uwerenge kumathandiza kupeza ngodya zabwino za mu makona obtuse. Choncho, tikhoza kunena, zochokera Lingaliro Lovomerezeka pamwambazi Uwerenge kumathandiza kupeza ngodya zabwino ndi obtuse wa makona ndi madigiri 180. Kachiwiri, Lingaliro Lovomerezeka izi safuna kuti adzathe umboni.