Yuav ua li cas mus nrhiav tau qhov chaw ntawm lub quadrilateral?

Yog hais tias lub dav hlau tau xwm yeem kos ob peb theem li hais tias ib tug yuav tsum tau pib nyob rau lub point qhov twg lub yav dhau los ib tug lawm, peb tau ib tug tawg kab. Cov theem yog hu ua kev mus, thiab qhov chaw uas lawv tshuam - saum. Thaum lub kawg ntawm lub xeem ya tshuam thawj starting taw tes, peb nrhiav tau ib tug kaw lawm kab, uas cais lub dav hlau mus rau hauv ob qhov chaw. Ib tug ntawm lawv yog finite, thiab lub thib ob infinite.

Tej yam yooj yim kaw nkhaus nrog tus uas xa tuaj nrog ib feem ntawm ib tug dav hlau (uas yog finite) yog hu ua ib tug polygon. Cov theem yog ob tog, thiab cov ces kaum tsim los ntawm lawv - saum. Cov nab npawb ntawm cov sab ntawm tej polygon sib npaug zos rau cov xov tooj ntawm vertices. Ib tug daim duab uas muaj peb sab, hu ua ib tug daim duab peb sab, tab sis plaub - ib tug quadrilateral. Polygon txaus yus los ntawm xws li qhov ntau raws li qhov chaw uas qhia tau hais tias qhov luaj li cas ntawm daim duab. Yuav ua li cas mus nrhiav tau qhov chaw ntawm lub quadrilateral? Qhia los ntawm ib tug ceg ntawm kev kawm txog zauv - geometry.

Yuav nrhiav tau cov cheeb tsam ntawm ib tug quadrilateral, nws yog tsim nyog los paub dab tsi hom nws yog - convex los yog nonconvex? Convex polygon tag nrho yog tus ncaj (thiab nws yuav tsum muaj ib yam ntawm cov ob tog) nyob rau hauv tib lub sab. Tsis tas li ntawd, muaj cov hom ntawm quadrilaterals raws li ib tug parallelogram nrog daws sib npaug thiab thaum uas tig mus opposite sab (ntau yam nws duab plaub nrog ncaj fab, rhombus nrog sib npaug sab, square nrog tag nrho cov cai ces kaum thiab plaub sib npaug zos sab), Trapezoid nrog ob tug uas tig mus opposite sab thiab deltoid nrog ob khub ntawm ib sab sab yog muaj sib npaug.

Squares tej polygon siv ib qho qauv, uas yog mus ua txhaum nws mus rau hauv cov vajvoos peb sab, txhua daim duab peb sab xam arbitrary cheeb tsam thiab quav cov tau. Tej convex quadrilateral yog muab faib ua ob peb ceg, nonconvex - ob los yog peb ntawm daim duab peb sab, lub cheeb tsam ntawm nws nyob rau hauv cov ntaub ntawv no tej zaum yuav muaj xws li cov sum thiab sib txawv ntawm cov kev tshwm sim. Lub cheeb tsam ntawm tej daim duab peb sab yog xam raws li ib nrab ntawm lub hauv paus cov khoom ntawm (a) lub siab (h), nqa tawm rau lub hauv paus. Cov mis uas yog siv nyob rau hauv cov ntaub ntawv no rau lub xam yog sau raws li: S = ½ • ib tug • h.

Yuav ua li cas mus nrhiav tau qhov chaw ntawm ib tug quadrilateral, piv txwv li, ib tug parallelogram? Nws yog tsim nyog los paub qhov ntev ntawm lub hauv paus (a), ib tug sab ntev (ƀ) thiab nrhiav tau cov sine ntawm lub kaum sab xis α, tsim los ntawm lub hauv paus thiab lub sab (sinα), rau xam cov mis yog li: S = ib tug • ƀ • sinα. Txij li thaum lub sine ntawm lub kaum sab xis α yog qhov khoom ntawm ib lub hauv paus ntawm ib tug parallelogram rau nws qhov siab (h = ƀ) - ib tug kab perpendicular mus rau lub hauv paus, nws cheeb tsam yuav muab xam rau qhov siab ntawm nws puag: S = ib tug • h. Los mus laij lub cheeb tsam ntawm ib tug rhombus thiab ib tug duab plaub kuj fits cov mis no. Txij li thaum lub sab sab ntawm lub duab plaub coincides nrog rau qhov siab ƀ h, nws cheeb tsam yog xam los ntawm cov mis S = ib tug • ƀ. Lub cheeb tsam ntawm lub square, vim hais tias ib tug = ƀ, yuav ua ncaj rau cov square ntawm nws cov sab: S = ib • a = a² . Lub cheeb tsam ntawm lub Trapezoid yog xam raws li ib nrab ntawm cov sum ntawm nws sab, multiplied by lub qhov siab (nws yog ua rau lub hauv paus ntawm lub Trapezoid perpendicular mus rau): S = ½ • (a + ƀ) • h.

Yuav ua li cas mus nrhiav tau qhov chaw ntawm lub quadrangle, Yog hais tias tsis paub hais tias ntev ntawm nws sab, tab sis yog paub rau nws kab pheeb ces kaum (e) thiab (f), thiab cov sine ntawm lub kaum sab xis α? Nyob rau hauv cov ntaub ntawv no qhov chaw ntawd yog xam raws li ib nrab ntawm cov khoom ntawm nws cov diagonals (cov kab uas txuas lub vertices ntawm lub polygon), multiplied los ntawm cov sine ntawm lub kaum sab xis α. Cov mis yuav tsum muab sau nyob rau hauv daim ntawv no: S = ½ • (e • f) • sinα. Nyob rau hauv kev rhombus cheeb tsam nyob rau hauv rooj plaub no yuav tsum tau sib npaug zos rau ib nrab ntawm cov khoom ntawm tus diagonals (cov kab sib txuas opposite fab ntawm ib tug rhombus): S = ½ • (e • f).

Yuav ua li cas mus nrhiav tau qhov chaw ntawm ib tug quadrilateral, uas tsis yog ib tug parallelogram los yog ib tug Trapezoid, nws yog feem ntau xa mus rau raws li ib tug arbitrary duab plaub. Lub cheeb tsam ntawm daim duab qhia nyob rau hauv cov nqe lus ntawm nws ib nrab-puag ncig (Ρ - lub sum ntawm ob sab nrog ib qho kab los sib ntsib), lub sab ib tug, ƀ, c, d, thiab cov sum ntawm ob opposite ces kaum (α + β): S = √ [(Ρ - a) • (Ρ - ƀ) • (Ρ - c) • (Ρ - d) - ib tug • ƀ • c • d • cos² ½ (α + β)].

Yog hais tias quadrilateral inscribed nyob rau hauv ib lub voj voog, thiab φ = 180 °, nyob rau hauv thiaj li yuav xam nws cov cheeb tsam siv Brahmagupta mis (Indian astronomer thiab mathematician, uas nyob rau hauv 6-7 centuries AD): S = √ [(Ρ - a) • (Ρ - ƀ) • (Ρ - c) • (Ρ - d)]. Yog hais tias quadrilateral piav ncig, ces (a + c = ƀ + d), thiab nws cheeb tsam yog xam: S = √ [ib tug • ƀ • c • d] • kev txhaum ½ (α + β). Yog hais tias lub quadrangle yog ib txhij piav ib voj voog thiab lub inscribed vajvoog mus rau lwm yam, lub cheeb tsam siv los laij cov nram qab no mis: S = √ [ib tug • ƀ • c • d].