Verengai pamakona ose netriangle. The theorem no musi Verengai pamakona ose netriangle

The netriangle iri polygon rine mativi matatu (pamakona matatu). Kakawanda, chikamu runotaurwa tsamba duku tsamba nemabhii makuru, inomiririra akatarisana vertices. Munyaya ino isu kutora Kutarira kwemarudzi aya geometric zvitarisiko, theorem no, iyo inorondedzera chii akaenzana uwandu pamakona ose netriangle.

Types pamakona pakukura

Zvinotevera zvokuzvivaraidza polygon pamwe vertices matatu:

• chenjera-angled, umo vose pamakona yakapinza;
• rectangular kuva mumwe kurudyi yorusvingo, parutivi aindiumba payo, akataura makumbo, uye parutivi kuti unongoda pakatarisana kurudyi yorusvingo anonzi hypotenuse;
• obtuse kana mumwe angled ndiye obtuse ;
• isosceles, ane mativi maviri vakaenzana, uye dzinonzi lateral, uye wechitatu - netriangle ane chawo;
• equilateral kuva vakaenzana kumativi matatu.

ehupfumi

Govera dzinokosha ehupfumi kuti vari yo mumwe mhando Triangle:

• pakatarisana chikuru divi iri angled nguva dzose mukuru, uye vakutarisire;
• vakaenzana pamakona pakatarisana akaenzana pakukura nebato, ndiEzekieri;
• chero netriangle ine mbiri chakasimba pamakona;
• rokunze angled kupfuura womukati angled asina pedyo kwazviri;
• Verengai chero maviri pamakona anogara asingasviki 180 madhigirii;
• kunze angled vakaenzana Verengai mamwe maviri pamakona, izvo zvisina mezhuyut naye.

The theorem no musi Verengai pamakona ose netriangle

The theorem no rinoti kuti kana iwe wedzera zvose makona geometric chimiro, riri munzvimbo Euclidean mundege, ipapo mari yavo ichava 180 inegumi. Ngatiedzei aratidze theorem no ichi.

Regai tine zvemasanga netriangle pamwe vertices KMN. Mhiri pamusoro M achabatwa zvakananga akaererana mutsetse KN (kunyange mutsetse uyu anonzi Euclid). Zvinofanira kucherechedzwa nguva A zvokuti pfungwa K uye A vanosarudzirwa kubva kumativi akasiyana pamusoro mutsetse MN. Tinowana chete kwekona pamusoro AMS uye MUF, izvo, kufanana mukati, nhema crosswise ndokuumba intersecting MN pachiimbwa zvakananga CN uye MA, izvo zvinofanana. Kubva ichi zvinoreva kuti vakaverenga pamakona ose Triangle, riri pamukova vertices pakati M uye N akaenzana ukuru CMA yorusvingo. All zvitatu pamakona ine Nemutengo akaenzana Verengai pamakona ose KMA uye MCS. Sezvo mashoko vari munyika pamakona hama mativi inofambirana mitsetse CL uye CM MA pa intersecting, mari yavo 180 inegumi. Izvi zvinoratidza kuti theorem no.

mugumisiro

Pakati pamusoro apa theorem no zvinoreva zvinotevera corollary: netriangle ose ane mbiri chakasimba pamakona. Kuti aratidze izvi, ngatiregei kufunga kuti ichi geometrical nhamba ane chete rikwanise angled. Unogona zvakare kufunga kuti hapana pamakona havasi unopinza. Muchiitiko ichi anofanira kuva pamakona zviviri, kwakakura iyo akaenzana kana kupfuura madhigirii 90. Asi zvino Verengai pamakona mukuru 180 inegumi. Asi izvi hazvigoni kuva, sezvo maererano theorem no kuwanda pamakona ose netriangle akaenzana 180 ° - haachatongondipinduri, kwete zvishoma. Ndizvo zvaifanira kwakaratidza.

Property kunze makona

Chii vakaverenga pamakona ose netriangle, zviri kunze? Mhinduro yomubvunzo uyu inogona kuwanikwa nokushandisa imwe nzira mbiri. Chokutanga ndechokuti unofanira kuwana Verengai pamakona, ayo akatorwa mumwe nomumwe vertex, kureva, vatatu pamakona. Yechipiri chinoreva kuti unofanira kuwana Verengai pamakona matanhatu panguva vertices. Kugadzirisa pakutanga wokutanga musimboti. Nokudaro, Triangle rine pamakona nhanhatu yokunze - panguva peimwe neimwe zviviri. peya imwe neimwe ine yakaenzana pamakona pakati pachavo, sezvo vari kutwasuka:

∟1 = ∟4, ∟2 = ∟5, ∟3 = ∟6.

Uyezve, zvinozivikanwa kuti rokunze pakona netriangle vakaenzana Verengai maviri mukati, izvo vasiri mezhuyutsya naye. saka,

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

Kubva ichi zvinoratidzika kuti Verengai panze pamakona, ayo akatorwa mumwe nomumwe pedyo nomumwe vertex achava akaenzana:

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

Akapiwa chokwadi kuti Verengai pamakona vakaenzana 180 madhigirii, zvinganzi nharo kuti ∟A + ∟V ∟S = + 180 °. Izvi zvinoreva kuti ∟1 + ∟2 + ∟3 = 2 X 180 ° = 360 °. Kana kusarudza wechipiri rinoshandiswa, Verengai pamakona matanhatu achava akagamuchirawo mukuru kaviri. Kureva vakaverenga pamakona ose netriangle kunze achava:

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

kurudyi Triangle

Chii akaenzana vakaverenga pamakona ose netriangle rworudyi, ndicho chitsuwa? Mhinduro, zvakare, kubva Theorem, iyo inoti kuti pamakona ose netriangle wedzera kusvika 180 inegumi. A inzwi kutaura kwedu (pfuma) sezvinotevera: mune netriangle zvakanaka unopinza pamakona wedzera kusvika madhigirii 90. Tinoratidza veracity kwayo. Ngakuve kupiwa netriangle KMN, iyo ∟N = 90 °. Zvakakodzera kuratidza kuti ° ∟K ∟M = + 90.

Saka, maererano theorem no musi Verengai pamakona ∟K + ∟M ∟N + = 180 °. Mumamiriro ezvinhu aya zvinonzi ∟N = 90 °. Zvakaratidza ∟K ∟M + + 90 ° = 180 °. Ndicho ∟K ∟M + = 180 ° - 90 ° = 90 °. Ndizvo tinofanira kuratidza.

Kuwedzera pamusoro apa ehupfumi ose netriangle zvakanaka, unogona kuwedzera izvi:

• pamakona, iro nhema makumbo yakapinza;
• ari hypotenuse of mativi mukuru kupfuura makumbo;
• Verengai makumbo kupfuura hypotenuse;
• cherwendo Triangle, riri pakatarisana pakona rworwendo 30, hafu hypotenuse, kuti yakaenzana hafu yayo.

Somumwe pfuma yacho geometric chimiro zvinogona kupatsanurwa Pythagorean theorem no. Iye anoti ari netriangle pamwe necheparutivi rworwendo 90 (rectangular), vakaverenga dzinoungana vanhu makumbo vakaenzana muchivara yacho hypotenuse.

Verengai pamakona ane isosceles Triangle

Pakuvamba takati kuti netriangle isosceles chinhu polygon pamwe vertices matatu, ine kuenzana mativi maviri. pfuma iyi inozivikanwa geometrical mufananidzo: pamakona panguva yayo vakaenzana. Ngationei zvinoratidza izvi.

Tora Triangle KMN, riri isosceles, SC - chawo. Tinosungirwa kuratidza kuti ∟K = ∟N. Saka, ngatirambei vanofungidzira kuti MA - KMN ndiye bisector kuti Triangle yedu. ICA Triangle chete chiratidzo wakanyatsojeka rokutanga Triangle MNA. Kureva, nokuda pfungwa yakapiwa kuti CM = NM, MA vanowanzochema divi, ∟1 = ∟2, nekuti MA - bisector ichi. Achishandisa wakaenzana pakati zvokuridza vaviri, mumwe aigona nharo kuti ∟K = ∟N. Nokudaro, theorem no kunoratidzwa.

Asi isu tiri kuziva, chii vakaverenga pamakona ose netriangle (isosceles). Nokuti iyi haina mamiriro ayo, tichava kutanga kubva theorem no inokurukurwa kare. Ndiko, tinogona kutaura kuti ∟K + ∟M ∟N + = 180 °, kana 2 X ∟K ∟M + = 180 ° (sezvo ∟K = ∟N). Izvi zvichaita kwete kuratidza pfuma, sezvo theorem no musi vakaverenga pamakona ose netriangle kwakaratidzei ambotaurwa.

Kunze vaiona ehupfumi ari makona netriangle, panewo vakadaro mashoko anokosha:

• mune imwe equilateral netriangle urefu, yakanga akatura kuna nechigadziko, ndiyo imwe chete muMedhia akapiwa bisector ari pakona riri pakati akaenzana kumativi uye akabatana kuti symmetry ose chawo;
• evaMedhia (bisector, kukwira), inoitiswa kumativi ari geometric nechipi, vakaenzana.

equilateral Triangle

It rinonziwo rworudyi, ndiye Triangle, izvo zvakaenzana kune mapato. Uye naizvozvo zvakaenzana uye pamakona. Mumwe nomumwe wavo 60 inegumi. Ngationei ratidza pfuma iyi.

Ngationei kufunga kuti tine netriangle KMN. Tinoziva kuti KM = HM = KH. Izvi zvinoreva kuti, maererano enzvimbo pamakona riri mujinga muimba equilateral netriangle ∟K = ∟M = ∟N. Sezvo, maererano Verengai pamakona ose netriangle theorem no ∟K + ∟M ∟N + = 180 °, ipapo x 3 = 180 ° ∟K kana ∟K = 60 °, ∟M = 60 °, ∟N = 60 °. Saka, kutaura kunoratidzwa. Sezvinoonekwa kubva apa uchapupu inobva kumusoro theorem no, vakaverenga pamakona ane equilateral Triangle, sezvo vakaverenga pamakona imwe netriangle iri 180 inegumi. Zvakare kuratidza theorem no ichi hachisi zvakakodzera.

Kuchine vamwe ehupfumi yo ane equilateral Triangle:

• evaMedhia bisector urefu mune geometrical mufananidzo akafanana, uye avo urefu rinowanikwa se (a X √3): 2;
• kana polygon ichi circumscribing denderedzwa, ipapo nharaunda achava akaenzana (a X √3): 3;
• kana nokudhindwa denderedzwa equilateral, triangle, ayo nharaunda yaizova (a X √3): 6;
• nzvimbo geometric nhamba rinowanikwa kubudikidza nzira: (A2 X √3): 4.

Obtuse Triangle

By tsanangudzo, imwe netriangle obtuse-angled, mumwe pamakona ayo ari pakati 90 kusvika 180 inegumi. Asi akapiwa chokwadi kuti vamwe vaviri pamakona ose geometric chimiro unopinza, zvinganzi vakagumisa kuti haapfuuri madhigirii 90. Naizvozvo, vakaverenga pamakona ose netriangle theorem no anoshanda pakuverenga Verengai pamakona muimba obtuse Triangle. Saka, tinogona zvakachengeteka kutaura, inobva kumusoro theorem no kuti vakaverenga obtuse pamakona ose netriangle iri 180 inegumi. Zvakare, theorem no ichi haadi kuzodzosera uchapupu.