Indlela yokuthola indawo quadrilateral?

Uma indiza, alokhu ukudweba izingxenye ezihlukahlukene ukuze kuthi uma omunye kufanele uqale endaweni lapho lowo odlule iphelile, sithola umugqa eziphukile. Lezi izingxenye kukhona okuthiwa izixhumanisi, kanye nezindawo lapho aphambana - eziqongweni. Lapho ukuphela ingxenye yokugcina ephambana ngowokuqala yekucala, sithola ivaliwe umugqa eziphukile, okuyinto uhlukanisa indiza yaba izingxenye ezimbili. Omunye wabo ezilinganiselwe, kanti eyesibili elingapheli.

Simple ijika avaliwe ingxenye ebiyelwe indiza (ukuthi okuyinto ezilinganiselwe) ibizwa ngokuthi ipholigoni. Amazenge abanengxenye, kanye nama-engeli ezakhiwe kubo - temabhodlela. Isibalo izinhlangothi ipholigoni ilingana nenani vertices. Isibalo esesikhundleni izinhlangothi ezintathu, okuthiwa unxantathu, kodwa amane - a quadrilateral. Polygon anda libhekene ezidlula ezifana endaweni elibonisa Usayizi sibalo. Indlela yokuthola indawo quadrilateral? Ukufundiswa igatsha wezibalo - izibalo zama-engela.

Ukuze uthole indawo quadrilateral, kubalulekile ukwazi ukuthi hlobo kungokwalabo - convex noma nonconvex? Convex ipholigoni lonke kuqhathaniswa ngqo (futhi kumele iqukathe yimiphi amaqembu) ohlangothini olufanayo. Ngaphezu kwalokho, kukhona izinhlobo quadrilaterals njengoba parallelogram nge Okusizayo alinganayo kanye parallel izinhlangothi eziphikisanayo (ezihlukahlukene kuye isikwele nge emakhoneni iqonde, rhombus nge amacala alingane, sikwele nazo zonke engele kwesokudla futhi izinhlangothi ezine ezilinganayo), trapezoid nge parallel amabili ezinhlangothini ezihlukene futhi deltoid ne amabili namabhuzu izinhlangothi eduze bayalingana.

Izikwele iyiphi ipholigoni basebenzisa indlela evamile, okuyinto ukuwuchitha ku onxantathu, unxantathu ngamunye ukubala indawo ngokungenasizathu bese ugoba le miphumela. Noma yimuphi quadrilateral ayizindunduma sehlukaniswe onxantathu ezimbili, nonconvex - ababili noma abathathu unxantathu, endaweni ke kulesi simo kungase kuhilele isamba kanye umehluko ngemiphumela. Le ndawo kwanoma iyiphi unxantathu esibaliwe kwesigamu umkhiqizo isizinda (a) ukuphakama (h), kwenziwe ukuze base. Ifomula esetshenziswa kulesi simo sokubalwa kulotshiwe njengoba: S = ½ • a • h.

Indlela yokuthola kwendawo quadrilateral, isibonelo, i-parallelogram? Kuyadingeka ukuba bazi ubude base (a), ubude eceleni (ƀ) bese uthola i-sine ye α engela, ezakhiwe base futhi ohlangothini (sinα), ukubala Ifomula njengoba: S = a • ƀ • sinα. Njengoba i-sine ye α engela iwumkhiqizo phansi a parallelogram ku ukuphakama kwalo (h = ƀ) - umugqa perpendicular isizinda, indawo yayo ibalwa ngokuvele azalane ukuphakama kwesisekelo salo: S = a • h. Ukuze abale kwendawo rhombus futhi isikwele iyazifanela lokhu ifomula. Njengoba uhlangothi lateral we isikwele kuvumelana H ukuphakama ƀ, endaweni yayo ibalwa ifomula S = a • ƀ. Le ndawo kwesikwele, ngoba = ƀ, kuyoba ulingana esigcawini uhlangothi lwalo: S = a • a = a² . Indawo trapezoid esibaliwe isigamu isamba ezinhlangothini zalo, iphindwe ukuphakama (iqhutshwa ukuba kumila trapezoid perpendicular): S = ½ • (a + ƀ) • h.

Indlela yokuthola indawo quadrangle, uma ubude obungaziwa ezinhlangothini zalo, kodwa liyaziwa idayagonali yayo (e) no (f), futhi sine ye-α engela? Kulokhu endaweni esibaliwe isigamu umkhiqizo diagonals yayo (imigqa ukuxhuma vertices the ipholigoni), iphindwe sine ye-α-engeli. Ifomula ungalotshwa leli fomu: S = ½ • (e • f) • sinα. Ikakhulukazi rhombus endaweni kulokhu kuyoba ulingana isigamu umkhiqizo diagonals (emigqeni yokuxhuma emakhoneni okuphambene a glossy): S = ½ • (e • f).

Indlela yokuthola kwendawo quadrilateral, engesiyo parallelogram noma trapezoid, sivame ukubizwa ngokuthi i isikwele ezingafanele. Le ndawo sibalo uzwakalise ngokuya yayo uhhafu-ipherimitha (Ρ - isamba izinhlangothi ezimbili nge-vertex ezivamile), izinhlangothi a, ƀ, c, d, futhi isibalo engeli amabili okuphambene (α + β): S = √ [(Ρ - a) • (Ρ - ƀ) • (Ρ - c) • (Ρ - d) - a • ƀ • c • d • cos² ½ (α + β)].

Uma quadrilateral alotshiwe embuthanweni, futhi φ = 180 °, ukuze ukubala endaweni yayo esetshenziswa Brahmagupta ifomula (sezinkanyezi Indian kanye sezibalo, owayehlala 6-7 eminyaka AD): S = √ [(Ρ - a) • (Ρ - ƀ) • (Ρ - c) • (Ρ - d)]. Uma quadrilateral echazwe selilonke ke (a + c = ƀ + d), kanye endaweni yayo ibalwa: S = √ [a • ƀ • c • d] • isono ½ (α + β). Uma quadrangle is kanyekanye wachaza omunye umbuthano kanye umbuthano elalibhalwe kuya komunye, endaweni esetshenziswa ukubala le ndlela elandelayo: S = √ [a • ƀ • c • d].