Quadrangle nrog txoj cai ces kaum - yog ib tug sum ntawm cov ces kaum ntawm ib tug quadrilateral ...

Ib qho ntawm feem nthuav xai nyob rau hauv lub geometry ntawm lub tsev kawm ntawv xyoo no - yog "ib tug quadrilateral" (Qib 8). Yuav ua li cas hom ntawm cov nuj nqis tshwm sim, dab tsi tshwj xeeb zoo lawv muaj? Yuav ua li cas yog nws hais txog quadrangles nrog cov ces kaum ntawm cuaj caum degrees? Cia saib tag nrho cov no.

Yuav ua li cas geometrical daim duab hu ua ib tug quadrangle

Uas hais tias muaj plaub sab, ntsig txog, ntawm cov plaub vertices (fab) yog hu ua nyob rau hauv Euclidean geometry quadrangles.

Xav nyob rau hauv lub keeb kwm ntawm no hom ntawm lub npe cov nuj nqis. Nyob rau hauv lub Lavxias teb sab lus lub noun "quadrilateral" yog muab los ntawm lub "plaub fab" nqe lus (nyob rau hauv tib txoj kev raws li cov "daim duab peb sab" - peb cov ces kaum, "pentagon" - tsib ces kaum, thiab lwm yam ...).

Txawm li cas los, nyob rau hauv Latin (uas tuaj los ntawm qhov kev sib kho uas muaj ntau geometric cov nqe lus nyob rau hauv feem ntau hom lus ntawm lub ntiaj teb no) nws yog hu ua ib tug quadrilateral. Lo lus no yog ib tug tus nab npawj Quadri (plaub) thiab ib tus latus (sab). Yog li ntawd peb yuav xaus uas lub ancient no polygon twb paub tsuas yog raws li "quadrilateral".

Los ntawm txoj kev, lub npe (nrog tus haum rau cov muaj cov nuj nqis ntawm no hom ntawm plaub sab, tsis fab) muab khaws cia tseg nyob rau hauv ib co niaj hnub lus. Piv txwv li, nyob rau hauv lus Askiv - quadrilateral thiab nyob rau hauv Fabkis - quadrilatère.

Nyob rau hauv feem ntau Slavic lus no tsiaj yog pom tias cov nuj nqis tseem nyob rau hauv lub xov tooj ntawm cov ces kaum, tsis lub sab. Piv txwv li, nyob rau hauv lub Slovak (štvoruholník), nyob rau hauv lub Bulgarian ( 'chetiriglnik ") nyob rau hauv Belarus (" chatyrohkutnіk ") nyob rau hauv Ukrainian (" chotirikutnik "), nyob rau hauv Czech (čtyřúhelník), tab sis nyob rau hauv lub Polish quadrangle hu ua nyob rau hauv lub xov tooj los ntawm ob tog - czworoboczny.

Yuav ua li cas hom ntawm quads tau kawm nyob rau hauv lub tsev kawm ntawv cov ntaub ntawv kawm

Nyob rau hauv niaj hnub geometry yog 4 hom uas nrog plaub sab. Txawm li cas los, vim rau heev thaj chaw ntawm ib co ntawm lawv nyob rau lub tsev kawm ntawv geometry kawm ntawv tsuas yog me ntsis txog cov ob hom.

• Parallelogram (parallelogram). Lub opposed sab ntawm lub quadrilateral yog thaum uas tig mus rau txhua lwm yam thiab, ntsig txog, yog muaj sib npaug nyob rau hauv officers.
• Trapezoid (trapezium los yog Trapezoid). Qhov no quadrilateral muaj ob opposite sab thaum uas tig mus txhua lwm yam. Txawm li cas los, lwm tus khub ntawm sab twb tsis muaj xws feature.

Tsis tau kawm nyob rau hauv lub tsev kawm ntawv chav kawm ntawm geometry hom quadrangles

Nyob rau hauv tas li ntawd rau tej no, muaj ob hom ntawm quadrangles uas cov menyuam kawm ntawv tsis paub mus rau lub geometry zaj lus qhia, vim hais tias ntawm lawv cov tshwj xeeb complexity.

• Deltoid (kite) - daim duab, nyob rau txhua lub ob khub ntawm ib sab sab sib npaug zos nyob rau hauv ntev rau txhua lwm yam. Lub npe ntawm no quadrangle yog vim lub fact tias nyob rau hauv tsos nws yog heev reminiscent ntawm cov tsab ntawv ntawm lub Greek cov tsiaj ntawv - "delta".
• Parallelogram (antiparallelogram) - qhov no daim duab yog li complex raws li nws lub npe. Nyob rau hauv nws ob lub opposite sab yog muaj sib npaug, tab sis lawv tsis thaum uas tig mus txhua lwm yam. Tsis tas li ntawd, qhov ntev ob tog hauv quadrangle tshuam raws li txuas ntxiv lwm yam ob luv luv sab.

hom parallelogram

Thaum hais txog nrog lub ntsiab hom quads, koj yuav tsum tau them mloog kom nws subspecies. Yog li ntawd, tag nrho cov parallelograms, nyob rau hauv lem, yog tseem muab faib ua plaub pawg.

• Classic parallelogram.
• Rhombus (rhombus) - quadrangular zoo sib npaug zos sab. Nws diagonals tshuam ntawm txoj cai ces kaum, faib lub rhombus ua plaub sib npaug zos txoj cai-angled cov vajvoos peb sab.
• Duab plaub (duab plaub). Qhov no lub npe hais lus rau nws tus kheej. Vim qhov no duab plaub nrog txoj cai ces kaum (txhua yam uas lawv sib npaug zos rau cuaj caum degrees). opposite sab tsis tau tsuas yog thaum uas tig mus txhua lwm yam, tab sis sib npaug zos.
• Square (square). Raws li tus duab plaub yog ib tug quadrilateral nrog txoj cai cov ces kaum, tab sis nws muaj tag nrho cov sab sib npaug. Qhov no, qhov no daim duab yog nyob ze rau ib tug pob zeb diamond. Yog li ntawd nws muaj peev xwm yuav sib cav hais tias lub square - yog ib tug ntoo khaub lig ntawm ib pob zeb diamond thiab ib tug duab plaub.

Cov khoom tshwj xeeb ntawm tus duab plaub

Xav cov nuj nqis, nyob rau hauv uas txhua txhua tus ntawm cov ces kaum ntawm lub sab yog sib npaug zos rau cuaj caum degrees, nws yog ib tug nqi los ze zog kom pom tseeb rau hauv lub duab plaub. Yog li ntawd, yog dab tsi nta nws muaj cov yam ntxwv uas txawv los ntawm lwm parallelograms?

Paub tias cov kev kawm parallelogram - ib tug duab plaub, nws diagonals yuav tsum sib npaug zos rau ib leeg, thiab txhua tus ntawm cov fab - ncaj. Nyob rau hauv tas li ntawd, lub square ntawm nws cov diagonals yuav tsum tau raws li cov sum ntawm cov plaub fab ntawm ob lub nyob ib sab sab ntawm lub duab. Nyob rau hauv lwm yam lus, lub classical duab plaub muaj ob txoj cai-angled cov vajvoos peb sab, raws li lawv yog hu, lub sum ntawm cov plaub fab ntawm ob txhais ceg yog sib npaug zos rau cov square ntawm lub hypotenuse. Nyob rau hauv lub luag hauj lwm ntawm lub hypotenuse ua hauj lwm pab kab pheeb ces kaum xam tau tias yog quadrangle.

Qhov kawg ntawm cov cim ntawm no daim duab tseem yog nws tshwj xeeb cov cuab yeej. Nyob rau hauv tas li ntawd, muaj lwm tus neeg. Piv txwv li, lub fact tias tag nrho ob tog kawm quadrangle nrog txoj cai ces kaum - yog ob qho tib si nws qhov siab.

Tsis tas li ntawd, Yog hais tias ib tug duab plaub nyob ib ncig ntawm txhua kos ib lub voj voog, nws txoj kab uas hla yuav sib npaug zos rau cov kab pheeb ces kaum ntawm lub inscribed nrhiav.

Cov lwm thaj chaw ntawm lub quadrilateral, qhov tseeb hais tias nws yog tiaj tus thiab uas tsis yog-Euclidean geometry tsis muaj nyob. Qhov no yog vim lub fact tias nyob rau hauv xws li ib tug system yog tsis muaj quadrangular daim duab, cov sum ntawm cov ces kaum yog sib npaug zos rau peb puas thiab rau caum degrees.

Lub square thiab nws cov nta

Thaum hais txog nrog tus yam ntxwv thiab thaj chaw ntawm lub duab plaub, koj yuav tsum xyuam xim rau lub thib ob lub npe hu science quadrangle nrog txoj cai ces kaum (a square).

Raws li nyob rau hauv qhov tseeb tib yam duab plaub, tab sis, nrog sib npaug zos sab, qhov no zoo muaj tag nrho cov nws cov khoom. Tab sis tsis zoo li nws, lub square yog tam sim no nyob rau hauv qhov uas tsis yog-Euclidean geometry.

Nyob rau hauv tas li ntawd, nyob rau hauv no daim duab, muaj lwm tus neeg tus yam ntxwv. Piv txwv li, qhov tseeb hais tias lub kab pheeb ces kaum ntawm ib tug square yog tsis tsuas sib npaug zos rau txhua lwm yam, tab sis tshuam ntawm txoj cai ces kaum. Yog li, raws li ib tug rhombus, ib tug square muaj plaub txoj cai-angled cov vajvoos peb sab, uas nws yog muab faib diagonally.

Nyob rau hauv tas li ntawd, qhov no daim duab yog lub feem ntau kuas ntawm tag nrho cov quadrangles.

Yuav ua li cas yog lub sum ntawm cov ces kaum ntawm ib tug quadrilateral

Xav tau cov yam ntxwv ntawm cov quadrangles ntawm Euclidean geometry, koj yuav tsum tau them sai sai rau lawv cov ces kaum.

Yog li, nyob rau hauv txhua tus ntawm cov saum toj no cov nuj nqis, txawm yog yus seb puas muaj nyob rau hauv nws txoj kev ces kaum los yog tsis tau, tus nqi ntawm lawv yeej tib yam nkaus - peb puas thiab rau caum degrees. Qhov no yog ib tug tshwj xeeb feature ntawm no hom ntawm cov nuj nqis.

puag ncig quadrangles

Thaum hais txog nrog hais tias, dab tsi yog qhov tawm ntawm cov ces kaum ntawm ib tug quadrilateral thiab lwm yam khoom tshwj xeeb ntawm cov duab ntawm no zoo, nws yog tsim nyog los paub tias dab tsi yog qhov zoo tshaj plaws los mus siv cov qauv los mus laij lawv puag ncig thiab thaj chaw.

Yuav kom txiav txim ntawm lub perimeter ntawm tej quadrilateral, xav tsuas ntxiv mus rau ib leeg qhov ntev ntawm nws sab.

Piv txwv li, nyob rau hauv Daim duab KLMN nws ncig yuav xam los ntawm cov mis: P = KL + LM + MN + KN. Yog hais tias peb hloov no tus xov tooj tau: 6 + 8 + 6 + 8 = 28 (cm).

Nyob rau hauv lub rooj plaub uas lub xam tau tias yog daim duab - ib tug square los yog ib tug rhombus, nrhiav ntawm lub perimeter ntawm lub mis yuav yooj yim zog los tsuas txawj qhov ntev ntawm ib qho ntawm nws cov sab ntawm plaub P x = KL piv txwv 4. 6 x 4 = 24 (cm).

Formula quadrangles Square

Thaum hais txog nrog yuav ua li cas nrhiav tau ntawm lub perimeter ntawm yam zoo nrog cov ces kaum plaub thiab sab yuav tsum xav txog cov feem ntau nrov thiab yooj yim txoj kev uas nrhiav tau nws cheeb tsam.

• Lub classic txoj kev uas yuav xam nws - qhov no yog siv cov mis S = 1/2 × haujlwm uas yog COy KM x SIN Lon. Nws hloov tawm hais tias tej cheeb tsam ntawm lub quadrangle yog sib npaug zos rau ib nrab ntawm cov khoom ntawm tus diagonals nyob rau sine ntawm lub kaum sab xis situated nruab nrab ntawm lawv.
• Yog hais tias daim duab uas nws cheeb tsam yuav tsum tau nrhiav - nws yog ib tug duab plaub los yog square (lub kab pheeb ces kaum uas yog ib txwm sib npaug zos rau txhua lwm yam), peb yuav piv cov mis, tsa nyob rau hauv lub square ntawm qhov ntev ntawm ib tug kab pheeb ces kaum thiab txawj nws los ntawm cov sine ntawm lub kaum sab xis ntawm lawv thiab faib nyob rau hauv ib nrab tag nrho. Piv txwv li: S = 1/2 CM ² x SIN Lon.
• Tsis tas li ntawd, thaum lub cheeb tsam ntawm ib tug duab plaub yuav pab tau txog cov puag ncig xam tau tias yog cov nuj nqis thiab qhov ntev ntawm ib qho ntawm nws cov sab. Nyob rau hauv xws li ib tug cov ntaub ntawv nws yuav ua tau tsim nyog siv cov mis S = KN x (P - 2 KN) / 2.
• Nyob rau hauv cov ntaub ntawv ntawm lub square ntawm nws cov khoom tso cai rau cov kev siv ntawm ob peb ntxiv qauv mus nrhiav qhov chaw ntawd. Piv txwv li, kev paub txog lub puag ncig nrhiav tej zaum yuav siv xws variant: S = P ² / 16. Thiab yog hais tias lub npe hu voos kheej-kheej ntawm lub inscribed lub voj voog rau ib tug quadrilateral, ib tug square cheeb tsam yog ib co zoo li txoj kev: S = 4r ^2. Yog hais tias lub voos kheej-kheej ntawm lub vajvoog yog lub npe hu, ces lwm tsim mis: S = 2R ^2. Tsis tas li ntawd, ib tug square cheeb tsam yog sib npaug zos rau 0.8 kab kos los ntawm lub ces kaum ntawm lub daim duab mus rau nruab nrab ntawm qhov opposite sab.
• Nyob rau hauv tas li ntawd rau tag nrho cov saum toj no, muaj kuj yog ib tug nyias muaj nyias ib cov mis mos rau nrhiav txoj kev hauv cheeb tsam, tsim los rau cov parallelogram. Nws yuav siv tau, yog hais tias paub, qhov ntev ntawm lub ob heights ntawm cov duab thiab qhov luaj li cas ntawm lub kaum sab xis ntawm lawv. Tom qab ntawd, qhov siab yuav tsum tau khoo nrog txhua lwm yam thiab lub sine ntawm lub kaum sab xis ntawm lawv. Nws yog ib nqi sau cia hais tias koj yuav siv cov mis no rau tag nrho cov nuj nqis, uas muaj feem rau cov parallelograms (ie, duab plaub, rhombus thiab square).

Lwm yam zog quadrangles: inscribed thiab circumscribed lub voj voog

Thaum pom tias tus yam ntxwv thiab cov khoom ntawm ib tug quadrangle li tus duab ntawm Euclidean geometry, yog tsim nyog hmoog rau lub tseem yuav piav qhia txog lub round los yog nkag mus rau hauv cov nram no:

• Yog hais tias tus lej ntawm qhov opposite ces kaum ntawm ib daim duab los ntawm ib puas thiab yim caum degrees thiab yog muaj sib npaug rau ib leeg, nws yog ua tau los piav qhia txog ib lub voj voog dawb do ib ncig ntawm no quadrangle.
• Raws li Ptolemy lub theorem, yog hais tias tus piav lub vajvoog sab nraum lub polygon nrog plaub sab, cov khoom ntawm lub diagonals sib npaug rau hauv lub sum ntawm cov khoom ntawm ob tog hauv daim duab. Yog li, cov mis yuav tsum: CM x haujlwm uas yog COy = KL x MN + LM x KN.
• Yog hais tias koj tsim ib cov duab plaub nyob rau hauv uas lub sum ntawm qhov opposite sab yog muaj sib npaug rau ib leeg, ces nws yog ua tau rau inscribe ib lub voj voog.

Thaum hais txog nrog lub fact tias xws li ib tug quadrilateral uas hom ntawm nws nyob ua ib ke, uas sawv daws yuav muaj xwb txoj cai ces kaum ntawm ob tog thiab yog dab tsi cov khoom lawv muaj, yuav tsum nco ntsoov tag nrho cov khoom no. Nyob rau hauv kev mis nrhiav puag ncig thiab thaj tsam ntawm lub polygons suav hais tias. Tom qab tag nrho, lub xam ntawm daim ntawv no - ib qho ntawm feem ntau, thiab txoj kev paub no yuav pab tau kom suav nyob rau hauv lub neej tiag tiag.