Balang tsa dikhutlo tsa triangolo ka. The Theorem ka chelete ea dikhutlo tsa triangolo ka

The triangolo ke khutlontsi le mahlakoreng a mararo (le dikhutlo e meraro). Hangata ka ho fetisisa, karolong e ka denoted ke litlhaku tse nyenyane ngollana litlhaku tse khōlō, a tšoantšetsang divertise bo fapaneng. A sehloohong sena a re hlahlobeng mefuta ena ya dibopeho tsa thutatekanyo, Theorem, e hlalosang mantsoe e hlalosa hore na ke ba lekanang le chelete ea dikhutlo tsa triangolo ka.

Mefuta dikhutloteng kholo ka ho fetisisa

Latelang mefuta khutlontsi le divertise tse tharo:

• hlobaetsang-angled, eo ho eona mahlakoreng 'ohle e bohale;
• likhutlo li 'nè le e mong hlaha ka lehlakoreng le letona, ka lehlakoreng le eaba ba etsa e le hlalositsweng maoto,' me lehlakoreng le hore e hanyetsa bo fapaneng ho hlaha ka lehlakoreng le letona o bitsoa hypotenuse;
• obtuse ha e mong hlaha ka lehlakoreng le e obtuse ;
• isosceles, eo mahlakoreng a mabeli a baa lekana, 'me li bitsoa lateral,' me ea boraro - ka triangolo le botlaaseng ka;
• matlhakore aa lekanang le tse tharo mahlakoreng lekanang.

thepa

Abelwang karolo ya ea thepa ea motheo tse bath ea e mong le mofuta oa triangolo:

• bo fapaneng lehlakoreng khōlō ka ho fetisisa e lula e khōloanyane angle, le sekgoeng;
• ke dikhutloteng lekanang mabapa le lekanang-kholo ka ho fetisisa oa mokha, le sekgoeng;
• ka triangolo efe kapa efe e na le dikhutlo e 'meli a hlobaetsang;
• angle tsa ka ntle a maholo ho feta leha e le hlaha ka lehlakoreng le ka hare se haufi tlhokeng;
• Balang tsa dikhutloteng efe kapa efe e 'meli e kamehla likhato tse ka tlaase ho 180;
• bokantle angle lekana chelete eo ea likhutlong tse ling tse peli tse, e leng ha ba mezhuyut le eena.

The Theorem ka chelete ea dikhutlo tsa triangolo ka

The Theorem bolela hore haeba u eketsa ho fihlela likhutlong tsohle tsa sebopeho thutatekanyo, eo e teng ka sefofane Euclidean, ka nako eo chelete ea bona e tla ba likhato tse 180. A re ke re leka ho bontša hore Theorem ena.

A re ke re be le triangolo hatellang le divertise KMN. Ka mose ho litlhōrō tsa M tla ahlola ntho e tšoanang ka ho toba ho ea moleng kn dostupnost (esita le mola ona o bitsoa Euclid). Re lokela ho hlokomela ntlha e le hore lintlha tse K le A ba a lokisetsa ho tswa ho mahlakoreng a fapaneng a lesika MN. Re fumana angle o tšoanang oa ho AMS le MUF, e leng, joaloka ka hare, a bua leshano crosswise ho theha intersecting MN ka kopanelo le CN tobileng le MA, tse tšoanang le. From ena e latelang e Balang tsa dikhutlo tsa triangolo e, teng ka divertise tsa M le N o lekana le boholo ba CMA angle. dikhutloteng tsohle tse tharo bopilwe ka chelete lekanang le chelete ea dikhutlo tsa KMA le MCS. Ho tloha ka ya data tse dikhutloteng hare a lekanyelitsoeng tsehelitseng tšoanang le mela CL le CM MA ka intersecting, chelete ea bona e ba likhato tse 180. Sena ke bopaki ba ho Theorem.

sephetho

Tse ka holimo Theorem ka holimo e fana ka maikutlo ho corollary latelang: mong le e mong triangolo na dikhutloteng tse peli a hlobaetsang. Ho paka sena, a re ke re nka hore palo ena jeometeri nang le 'ngoe feela a hlobaetsang le hlaha ka lehlakoreng le. U ka boela ua nka hore ha ho likhutlong le ha li bohale. Tabeng ena ee lokela ho ba mahlakoreng le bonyane ba babeli, boholo ba eo o lekana le kapa moholo ho feta likhato 90. Empa ka nako eo o bala dikhutloteng le o moholo ho feta likhato tse 180. Empa sena e ke ke, e le ho ea ka Theorem chelete dikhutlo tsa triangolo le o lekana le 180 ° - ha ho ho feta, ha ho ka tlaase ho moo. Ke seo ne a lokela ho pakoa.

Thepa kantle ho likhutlong

ke chelete ea dikhutlo tsa triangolo, e leng di a Link eng? The karabo ea potso ena ka fumanoa ka ho sebelisa e 'ngoe ea litsela tse peli. Ea pele ke hore u lokela ho fumana chelete ea dikhutloteng, e leng ba isoa e mong ka ho e mong le e vertex, ke hore, le dikhutlo tse tharo. Ea bobeli e bolela hore o lokela ho fumana chelete ea dikhutloteng tse tsheletseng ho divertise ena. Ho sebetsana le e le qalo ea ho mothofatso pele. Kahoo, triangolo na le tse tšeletseng tse likhutlong tsa ka ntle - ka holimo a ka 'ngoe ea tse peli. E mong le e para o na le dikhutlo e lekanang pakeng tsa bona, kaha ba ke paatsepama:

∟1 = ∟4, ∟2 = ∟5, ∟3 = ∟6.

Ho phaella moo, eo tsejoa hore ea sekhutlo le ka ntle la triangolo e lekana le chelete eo ba e 'meli ka hare, e leng se mezhuyutsya le eena. Ka hona,

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

From ena ho bonahala eka chelete eo ba dikhutloteng le bokantle, tse nkiloeng ka bonngoe haufi e mong le e vertex tla ho lekana le:

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

Fuoa 'nete ea hore chelete ya dikhutloteng ho lekana le likhato tse 180, e ka pheha khang ea hore ∟A + ∟V ∟S = + 180 °. Sena se bolela hore ∟1 + ∟2 + ∟3 = 2: x 180 ° = 360 °. Haeba khetho ea bobeli e sebediswa, Balang tsa dikhutloteng tse tsheletseng tla ka ho tšoanang e khōloanyane habeli. Ie Balang tsa dikhutlo tsa triangolo ka ntle e tla ba:

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

triangolo nepahetseng

Se lekanang le chelete ea dikhutloteng tsa triangolo letona, ke sehlekehleke sohle? Karabo ke, hape, ho tloha Theorem, e bolelang hore e le dikhutlo tsa triangolo e eketsa ho fihlela ho likhato tse 180. A nang le molumo polelo rona (thepa) e latelang: ka triangolo letona le dikhutlo e bohale eketsa ho fihlela ho likhato 90. Re bontša hore re veracity lona. Ho ke ho fuoa triangolo KMN, eo ∟N = 90 °. Ho ke ke ho hlokahala ho paka hore ∟K ∟M = + 90 °.

Kahoo, ho ea ka Theorem ka chelete ea dikhutloteng ∟K + ∟M ∟N + = 180 °. Li le boemong bo ena ho boleloa hore ∟N = 90 °. E fellang kateng le ∟K ∟M + + 90 ° = 180 °. Ke ∟K ∟M + = 180 ° - 90 ° = 90 °. Hore 's seo re neng re tšoanela ho paka.

Ho phaella ho thepa ka holimo tsa triangolo nepahetseng, o ka eketsa ana:

• dikhutloteng, eo leshano khahlanong le maoto le e bohale;
• le hypotenuse ea kgutlotharo e moholo ho e 'ngoe ea maoto a ka;
• Balang ea maoto ho feta ho feta hypotenuse ea;
• leoto la triangolo, e leng larileng bo fapaneng ho hlaha ka lehlakoreng le ya likhato tse 30, halofo ea hypotenuse, e le hore o lekana le halofo ea eona.

E le thepa e 'ngoe ea sebopeho thutatekanyo ka khetholloa Pythagorean Theorem. O ile a pheha khang ea hore ka triangolo ka hlaha ka lehlakoreng le ya likhato 90 (likhutlo li 'nè), chelete eo ea mapatlelong a maoto a ka lekana le ea lisekoere tsa hypotenuse ena.

Balang tsa dikhutloteng ya triangolo isosceles

Pejana ho moo re ile a bolela hore e triangolo isosceles ke khutlontsi le divertise tse tharo, e nang le tse peli mahlakoreng lekanang. thepa ena e tsebahalang tsa thutatekanyo palo: le dikhutlo ho botlaaseng lona lekanang. A re ke re bontše sena.

Nka triangolo KMN, e leng isosceles, SC - botlaaseng lona. Re lokela ho bontša hore ∟K = ∟N. Ho joalo, a re ke re nka hore MA - KMN ke bisector tsa triangolo rona. ICA triangolo le letšoao la pele la tekano ke triangolo MNA. E leng, ka khopolo fanoeng hore CM = NM, MA ke lehlakoreng tloaelehileng, ∟1 = ∟2, hobane MA - bisector ena. Sebedisa tekano ea kgutlotharo tse peli, e 'ngoe ka pheha khang ea hore ∟K = ∟N. Kahoo, Theorem bo pakoa.

Empa re na le thahasello ka, se Balang tsa dikhutloteng tsa triangolo (isosceles). Hobane ka tsela ena ha ho na le makgetha a ho lona, re tla qala ho tswa ho Theorem tšohloa pele. Ke hore, re ka re hore ∟K + ∟M ∟N + = 180 °, kapa 2: x ∟K ∟M + = 180 ° (jwalo ka ∟K = ∟N). Sena se tla hase bopaki ba thepa, e le ho Theorem ka chelete ea dikhutlo tsa triangolo e ile ipakile pejana.

Ntle le thepa e nkoa e sa likhutlong tsa triangolo, ho na le boetse ho na le lipolelo tse joalo tsa bohlokoa:

• ka e le matlhakore aa lekanang triangolo bophahamo, e neng e theolela ho botlaaseng, e ka nako ya bisector bohare ba hlaha ka lehlakoreng le leng ke pakeng tsa mahlakoreng lekanang le a selekane sa setshwani ya botlaaseng lona;
• bohare (bisector, bophahamong), tse neng e tšoaretsoe ho mahlakoreng a mokhabo thutatekanyo, baa lekana.

matlhakore aa lekanang triangolo

Ho e boetse e bitsoa e nepahetseng, ke triangolo, tse lekanang le mekga tsohle. 'Me ka hona hape e lekanang le dikhutlo. E mong le e ba bona ba e likhato tse 60. A re ke re paka thepa ena.

A re ke re nka hore re na le triangolo KMN. Rea tseba hore KM = HM = KH. Sena se bolela hore, ho ea ka thepa ea dikhutloteng le sebakeng botlaaseng ka matlhakore aa lekanang triangolo ∟K = ∟M = ∟N. Ho tloha, ho ea ka chelete ea dikhutloteng tsa triangolo Theorem ∟K + ∟M ∟N + = 180 °, ka nako eo: x 3 = 180 ° ∟K kapa ∟K = 60 °, ∟M = 60 °, ∟N = 60 °. Kahoo, polelo bo pakoa. Ka bone ho tloha ka bopaki ka holimo e thehiloeng Theorem tse ka holimo, Balang tsa dikhutloteng ya triangolo matlhakore aa lekanang, e le chelete ea dikhutlo tsa triangolo leha e le efe ke likhato tse 180. Hape le leke Theorem sena ha se ho hlokahala.

Ho na le ba ntse ba tse ling thepa bath ya triangolo matlhakore aa lekanang:

• bohare bisector bophahamo ka palo tsa thutatekanyo tšoanang, le bolelele ba bona ba e balwa ka (a x, √3): 2;
• haeba khutlontsi ena circumscribing lesakaneng, ka nako eo radius tla ho lekana le (a x, √3): 3;
• ha ho ngotsoe ka lesakaneng matlhakore aa lekanang triangolo, radius lona ka ba (a x, √3): 6;
• sebakeng sa palo thutatekanyo e balwa ka moralo wa: (a2: x √3): 4.

Obtuse triangolo

Ke tlhaloso, e obtuse-angled triangolo, e mong oa likhutlong tsa lona ke pakeng tsa likhato tse 90 ho tse 180. Empa fanoeng 'nete ea hore ba bang dikhutloteng tse peli tsa sebopeho thutatekanyo bohale, e ka etsa qeto ea hore ha ba feta likhato 90. Ka lebaka leo, chelete ea dikhutloteng tsa triangolo Theorem sebetsa ha ho baloa ho bala dikhutloteng le ka triangolo obtuse. Ho joalo, re ka ka tsela e sireletsehileng e re, e thehiloeng Theorem ka holimo hore ho bala dikhutloteng le obtuse tsa triangolo ke likhato tse 180. Hape, Theorem ena ha ho hlokahale hore boela bopaki.