Parallelogram vs Quadrilateral
Quadrilaterals thiab parallelograms yog polygons pom hauv Euclidean Geometry. Parallelogram yog ib qho tshwj xeeb ntawm cov plaub fab. Quadrilaterals tuaj yeem yog hom phiaj (2D) lossis 3 Dimensional thaum parallelograms yeej ib txwm npaj.
Quadrilateral
Quadrilateral yog ib lub voj voog nrog plaub sab. Nws muaj plaub vertices, thiab cov lej ntawm cov ces kaum sab hauv yog 3600 (2π rad). Quadrilaterals tau muab faib ua tus kheej sib tshuam thiab yooj yim quadrilateral pawg. Tus kheej-kev sib tshuam quadrilaterals muaj ob lossis ntau sab hla ib leeg, thiab cov duab geometric me me (xws li daim duab peb sab yog tsim nyob rau hauv lub quadrilateral).
Cov plaub hau yooj yim kuj tau muab faib ua convex thiab concave quadrilaterals. Concave quadrilaterals muaj ob sab uas nyob ib sab tsim reflex kaum sab hauv daim duab. Qhov yooj yim quadrilaterals uas tsis muaj lub kaum sab xis ntawm sab hauv yog convex quadrilaterals. Lub convex quadrilaterals ib txwm muaj tessellations.
Ib feem tseem ceeb ntawm geometry ntawm quadrilaterals ntawm cov theem pib cuam tshuam txog lub convex quadrilaterals. Ib txhia quadrilaterals yeej paub zoo rau peb los ntawm cov hnub ntawm cov tsev kawm ntawv theem pib. Hauv qab no yog daim duab qhia qhov sib txawv convex quadrilaterals.
Parallelogram
Parallelogram tuaj yeem txhais tau tias yog daim duab geometric nrog plaub sab, nrog rau sab nraud sib npaug rau ib leeg. Ntau precisely nws yog quadrilateral nrog ob khub ntawm parallel sab. Qhov xwm txheej sib npaug no muab ntau yam geometric yam ntxwv rau cov parallelograms.
Ib plaub plaub yog ib qho parallelogram yog tias pom cov yam ntxwv geometric.
• Ob khub ntawm kev tawm tsam yog sib npaug ntawm qhov ntev. (AB=DC, AD=BC)
• Ob khub ntawm lub kaum sab xis sib npaug. ([latex]D\hat{A}B=B\hat{C}D, A\hat{D}C=A\hat{B}C[/latex])
• Yog cov ces kaum uas nyob ib sab yog supplementary [latex]D\hat{A}B + A\hat{D}C=A\hat{D}C + B\hat{C}D=B\hat {C}D + A\hat{B}C=A\hat{B}C + D\hat{A}B=180^{circ}=\pi rad[/latex]
• Ib khub ntawm sab, uas tawm tsam ib leeg, yog sib npaug thiab sib npaug ntawm qhov ntev. (AB=DC & AB∥DC)
• Diagonals bisect ib leeg (AO=OC, BO=OD)
• Txhua kab kab pheeb ces kaum faib cov plaub fab rau hauv ob daim duab peb sab congruent. (∆ADB ≡ ∆BCD, ∆ABC ≡ ∆ADC)
Ntxiv mus, qhov sib npaug ntawm cov squares ntawm ob sab yog sib npaug rau cov lej ntawm cov squares ntawm kab pheeb ces kaum. Qhov no qee zaum hu ua txoj cai parallelogram thiab muaj kev siv dav hauv physics thiab engineering. (AB2 + BC2 + CD2 + DA2=AC2 + BD2)
Txhua tus yam ntxwv saum toj no tuaj yeem siv los ua cov khoom, thaum nws tau tsim tias lub plaub fab yog ib qho parallelogram.
Thaj tsam ntawm parallelogram tuaj yeem suav los ntawm cov khoom ntawm qhov ntev ntawm ib sab thiab qhov siab mus rau sab nraud. Yog li ntawd, thaj tsam ntawm parallelogram tuaj yeem teev tau raws li
Area of parallelogram=base × × h
Thaj tsam ntawm parallelogram yog ywj siab ntawm cov duab ntawm tus kheej parallelogram. Nws tsuas yog nyob ntawm qhov ntev ntawm lub hauv paus thiab qhov siab perpendicular.
Yog hais tias ob sab ntawm lub parallelogram tuaj yeem sawv cev los ntawm ob lub vectors, thaj chaw tuaj yeem tau txais los ntawm qhov loj ntawm cov khoom vector (khoom khoom hla) ntawm ob lub vectors uas nyob ib sab.
Yog ob sab AB thiab AD yog sawv cev los ntawm vectors ([latex]\overrightarrow{AB}[/latex]) thiab ([latex]\overrightarrow{AD}[/latex]) feem, thaj tsam ntawm parallelogram muab los ntawm [latex]\left | \overrightarrow{AB}\times \overrightarrow{AD} right |=AB\cdot AD \sin \alpha [/latex], where α is the angle between [latex]\overrightarrow{AB}[/latex] and [latex]\overrightarrow{AD}[/latex].
Ua raws li yog qee qhov khoom siab tshaj ntawm parallelogram;
• Thaj chaw ntawm ib qho parallelogram yog ob npaug ntawm thaj tsam ntawm daim duab peb sab tsim los ntawm ib qho ntawm nws kab pheeb ces kaum.
• Cheeb tsam ntawm qhov parallelogram muab faib ua ib nrab los ntawm ib kab twg dhau los ntawm qhov nruab nrab.
• Ib qho kev hloov pauv uas tsis yog degenerate affine yuav siv lub parallelogram mus rau lwm parallelogram
• Ib qho parallelogram muaj symmetry rotational symmetry 2
• Cov lej ntawm qhov kev ncua deb ntawm txhua qhov taw qhia sab hauv ntawm lub parallelogram rau ob sab yog ywj siab ntawm qhov chaw ntawm qhov taw tes
Qhov txawv ntawm Parallelogram thiab Quadrilateral yog dab tsi?
• Quadrilaterals yog polygons nrog plaub sab (qee zaus hu ua tetragons) thaum parallelogram yog hom tshwj xeeb ntawm plaub fab.
• Quadrilaterals tuaj yeem muaj lawv ob sab hauv cov dav hlau sib txawv (hauv 3d qhov chaw) thaum txhua sab ntawm lub parallelogram nyob rau tib lub dav hlau (planar / 2dimensional).
• Cov kaum sab hauv ntawm plaub lub kaum sab xis tuaj yeem coj tus nqi (nrog rau lub kaum ntse ntse) xws li lawv ntxiv txog 3600. Parallelograms tsuas muaj obtuse kaum sab xis raws li hom siab tshaj plaws ntawm lub kaum ntse ntse.
• Plaub sab ntawm cov plaub fab tuaj yeem sib txawv ntev thaum cov sab nraud ntawm qhov parallelogram ib txwm sib npaug rau ib leeg thiab sib npaug ntawm qhov ntev.
• Txhua kab kab pheeb ces kaum faib cov parallelogram ua ob daim duab peb sab sib dhos, thaum cov duab peb sab uas tsim los ntawm kab pheeb ces kaum ntawm cov plaub fab dav dav tsis tas yuav congruent.
