I ka huina o na pana pua o ka triangle. Ke theorem ma ka huina o ka pana pua o ka triangle

Ke triangle He He iieeaii me ekoluʻaoʻao (ekolu pana pua). Loa pinepine, i ka hapa denoted ma ka palapala uuku e like paukü palapala, a ho i ku pono vertices. Ma keia 'atikala, ua lawe i ka nana aku i kēia mauʻano o geometric kinona, theorem, i Ho'ākāka' ka mea like me ka huina o ka pana pua o ka triangle.

Ano nui loa pana pua

ʻO kēia mau 'ano o ka iieeaii me ekolu vertices:

• huʻi-huina, ma i ka pana pua a pau, Ua lena;
• mau huinahń loa me kekahi pono ka makau, i kaʻaoʻao ho okumu ia, haawiia keia i na wawae, a ua kapaʻia kaʻaoʻao i ka manaʻo e ku pono ana i ka huina akau o ka hypotenuse;
• obtuse ka wā kekahi huina mea obtuse ;
• isosceles, nona nāʻaoʻaoʻelua, ua like, a me ka poe i kaheaia kīloi paʻewa lakou, a me ke kolu - he triangle me ke kumu;
• equilateral me ekolu like aoao.

waiwai

Līkaia no ka kumu o ka waiwai i mea ano o kēlā me kēia 'ano o ka triangle:

• ku pono ana i kaʻaoʻao nui ka manawa a oi aku ka makau, a me ka hope versa;
• Ua kau like pana pua e ku pono ana i ka like-nui loa aoao, a me ka hope versa;
• ma kekahi triangle, he mau huʻi pana pua;
• mawaho huinaʻoi aku ma mua o kekahi na huina like ole e pili ana i;
• ka huina o kekahi mau pana pua, he mau emi ma mua o 180;
• paia waho huina lākou, i ka huina o na kihi'ē aʻeʻelua, i mea ole mezhuyut me ia.

Ke theorem ma ka huina o ka pana pua o ka triangle

Ke theorem hākālia nō ia ina e hookui aku i nā mea a pau i nā kihi o ka geometric kinona, i ua aia i loko o ka Euclidean pelane, a laila lākou huina e e 180. E ka ho'āʻo e hooiaio i kēia theorem.

E mākou i ka ākeʻakeʻa kumu triangle me vertices KMN. Ma ka luna o M, e paa i ka pololei ia laua a hiki i ka laina kn (i keia laina ua kapaia Euclid). It E e kaulana wahi A no laila, i nā mea nui a me K A e hooponopono, mai kekahiʻaoʻao o ka laina HI. Mākou kiʻi i ka huina o AMS a me MUF, a, e like me ka Kalaiaina, moe crosswise e lilo intersecting HI loko o ka huipūʻana me nā moʻopuna CN a me MA, i mea ia laua. Mai keia ka mea, penei i ka huina o na pana pua o ka triangle, Aia ma ka vertices o M a me N mea like me ka nui o ka CMA huina. A pau ekolu pana pua komo ana o ke dala e like me ka huina o ka pana pua o KMA a me MCS. No ka 'ikepili i na pana pua pili kapakahi laua laina Cl a me knm MA ma intersecting, kā lākou huina o 180. Kēia proves ka theorem.

hopena

O ka mea ma luna o ka luna theorem hoʻohuʻu ke kēia corollary: o kela mea keia triangle ua mau huʻi pana pua. E hoao i kēia, e mai kuhi i keia geometrical huahelu i hoʻokahi wale nō huʻi huina. E hiki no hoi kuhi i kekahi o na kihi e ole pahi. Ma keia hihia mea pono ia i ka liʻiliʻi mau pana pua, i ka nui o ka i mea like i paha i oi mamua o 90 degere. Akā, alaila, o ka huina o na pana pua, uaʻoi aku ma mua o 180. Akā, ua hiki ole e, like e like me ka theorem huina pana pua o ka triangle mea like me 180 ° - ole aku,ʻaʻohe emi. Ia ka leo o ka mea i ae e hoao.

Property ma waho kihi

He aha ka i ka huina o na pana pua o ka triangle, i mea mawaho? Ke pane i keia ninau hiki ke loaa e ana i kekahi o nāʻaoʻao. Ka mua ka mea e pono ai e imi i ka huina o ka pana pua, a i lawe kekahi ma kēlā me kēia vertex,ʻo ia hoʻi, ekolu pana pua. Ka lua o ka hoʻohuʻu ia oe pono e huli i ka huina o na pana pua eono ma na vertices. E hana me ka hoʻomaka o keʻano mua. Pela, ke triangle he eono waho kihi - ma ka luna o kela a me keia o na mau. Kēlā me kēia mau mea like pana pua ma waena o lakou iho, no ka mea, e vertical:

∟1 = ∟4, ∟2 = ∟5, ∟3 = ∟6.

Eia hou, ka mea, ua ikeia e lākou, i ka pā kihi o ka triangle i ka huina o ka mau Kalaiaina, i mea ole mezhuyutsya me ia. nolaila,

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

Mai keia eia kahi mea akaka i ka huina o ka paia waho pana pua, a i lawe kekahi ma kekahi kokoke i kēlā vertex e e like me:

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

Ua haawiia i ka mea e lākou, i ka huina o na pana pua 180, ka mea hiki ke ua kūkākūkāʻia ia ∟A + ∟V ∟S = + 180 °. Kēia 'o ia hoʻi i ∟1 + ∟2 + ∟3 = 2 m 180 ° = 360 °. Inā i ka lua o ka AOAIAaOO ua hoʻohana 'ia, i ka huina o na pana pua eono e e correspondingly oi aku i nā manawaʻelua. Ie o ka huina o na pana pua o ka triangle ma waho, e ia:

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

pono triangle

He aha mea like me ka huina o ka pana pua o ka triangle pono, o ka aina? Ke pane mea, hou, mai Theorem, i hākālia nō i ka pana pua o ka triangle hookui aku i ka 180. He walaʻau nō mākou assertion (waiwai) like penei: ma ka akau triangle oi pana pua hookui aku i ke 90 degere. E hoao kona veracity. E ka mea, e haawiia mai triangle KMN, a ∟N = 90 °. He mea pono e hooiaio i ∟K ∟M = + 90 °.

Penei, e like me ka theorem ma ka huina o na pana pua ∟K + ∟M ∟N + = 180 °. Ma keia ano ka mea, ua olelo mai ia ∟N = 90 °. Ua huli mai ∟K ∟M + + 90 ° = 180 °. I mea ∟K ∟M + = 180 ° - 90 ° = 90 °. O ia ka leo mea mākou pono, e hooiaio.

Ma waho aʻeo ia ka luna waiwai o ka triangle pono, e nui ana ke hooloihi i kēia mau:

• pana pua, a moe aku i na wawae, Ua lena;
• ka hypotenuse o ka triangular nui ma mua o kekahi o nā wāwae;
• i ka huina o nā wāwae oi ma mua o ka hypotenuse;
• uha mua o ka triangle, a moe e ku pono ana i ka huina o ka 30 degere, ka hapalua o ka hypotenuse, i mea like me kona hapalua.

E like me kekahi waiwai o ka geometric shape hiki e hookoaia Pythagorean theorem. Ua hoʻopaʻapaʻa i loko o ka triangle me ka huina o ka 90 degere (huinahń), lākou, i ka huina o na mika pāhoʻonui lua o ka wawae i ke kahua o ka hypotenuse.

I ka huina o ka pana pua o ka isosceles triangle

Mamua aku nei nō mākou i mai la oia i ka isosceles triangle mea he iieeaii me ekolu vertices, he mau likeʻaoʻaoʻelua. Keia waiwai ua ike geometrical huahelu: ka pana pua ma kona kumu like. E mākou hoao keia.

E lawe i ka triangle KMN, i mea isosceles, SC - kona kumu. e koi mākou e hooiaio ia ∟K = ∟N. No laila, e mai kuhi ia MA - KMN o ka bisector o ko mākou triangle. ICA triangle me ka hoailona mua o ka like o triangle MNA. Oia, ma ka kuhiakau haawiia i knm = HI, MA mea he mauʻaoʻao, ∟1 = ∟2, no ka mea, MA - keia bisector. E ho ohana i ka like o na triangle maunaʻelua, hoʻokahi hiki hoʻopaʻapaʻa ana i ∟K = ∟N. Nolaila, ke theorem ua hoao.

Akā, ke i hoihoi i loko, i mea o ka huina o na pana pua o ka triangle (isosceles). No ka mea, ma keia mau mea ia aʻole i kona hiʻona, e hoʻomaka mākou, mai ka theorem kūkākūkā mua. Ia mea, ua hiki ke olelo ia ∟K + ∟M ∟N + = 180 °, a 2 m ∟K ∟M + = 180 ° (me ∟K = ∟N). Kēia e hooiaio mai i ka waiwai, e like me ka theorem ma ka huina o na pana pua o ka triangle i hoao mamua.

Ina aole ka manaoia waiwai o na kihi o ka triangle, eia no hoi ia nui māmala'ōlelo e hōʻike ana:

• i loko o ka equilateral triangle kiʻekiʻe, a i ua kuu iho la i ka waihona ipu, o ka poʻe i ka Media bisector o ka huina i mea ma waena o ka like aoao, a me ka iho (axis) o symmetry o kona kumu;
• Median (bisector, kiʻekiʻena), a i paa ai i ka aoao o ka geometric huahelu, e like.

equilateral triangle

Ua ua i kapaʻia ai ka pono, o ka triangle, a e like me nā mea a pau aoao. A no ia mea, i like me ka pana pua. Kēlā me kēia no ia mea 60 degere. E mākou hoao i kēia waiwai.

E mai kuhi ana mākou i ke triangle KMN. Ua ike makou ua KM = HM = kh. Kēia 'o ia hoʻi ia, e like me ka waiwai o ka pana pua aia ma ka waihona ipu i loko o ka equilateral triangle ∟K = ∟M = ∟N. Mai, e like me ka huina o ka pana pua o ka triangle theorem ∟K + ∟M ∟N + = 180 °, laila m 3 = 180 ° ∟K a ∟K = 60 °, ∟M = 60 °, ∟N = 60 °. Pela, ke assertion ua hoao. Like ike mai i ka luna 'ölelo ma muli o ka luna theorem, i ka huina o na pana pua o ka equilateral triangle, e like me ka huina o ka pana pua o kekahi'ē aʻe triangle mea 180. Hou kona hoakaka keia theorem mea i pono.

Aia mau nō ka waiwai kekahi ano o ka equilateral triangle:

• Median bisector kiʻekiʻe i loko o ka geometrical aka'ālike, a me kā lākou lōʻihi he pōpilikia like (he m √3): 2;
• ina keia iieeaii circumscribing i ka p ÷ ai, alaila, o ka ke kahahńnai e e like ia (i ka m √3): 3;
• ina kākauʻia i loko o ka kaiapili equilateral triangle, kona kahahńnai makemake e (he m √3): 6;
• wahi o ka geometric huahelu he pōpilikia ma ka haʻilula: (A2 m √3): 4.

Obtuse triangle

Ma ka ho'ākāka 'ana, he obtuse-huina triangle, kekahi o kona mau kihi mea ma waena o 90 a hiki i 180. Akā, hāʻawi aku i ka mea ana i mau pana pua 'ē aʻe o ka geometric kinona pahi, ka mea hiki e manao ia ka mea e ole mamua o 90 degere. Nolaila, o ka huina o na pana pua o ka triangle theorem hana ma ka helu ana i ka huina o ka pana pua i loko o ka obtuse triangle. No laila, pakele olelo hiki mākou, ma muli o ka luna theorem i ka huina o ka obtuse pana pua o ka triangle mea 180. Hou, keia theorem aole i pono i ka hou 'ana i maopopo.