Parallelogram vs Rectangle
Parallelogram thiab duab plaub yog plaub fab. Lub geometry ntawm cov duab no tau paub rau txiv neej rau ntau txhiab xyoo. Cov ntsiab lus tau hais meej meej hauv phau ntawv "Elements" sau los ntawm Greek lej Euclid.
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
Rectangle
Ib lub plaub fab nrog plaub lub kaum sab xis yog hu ua lub duab plaub. Nws yog ib qho tshwj xeeb ntawm parallelogram qhov twg cov ces kaum ntawm ob sab uas nyob ib sab yog cov kaum sab xis.
Ntxiv rau tag nrho cov khoom ntawm ib qho parallelogram, cov yam ntxwv ntxiv tuaj yeem pom tau thaum xav txog cov duab plaub.
• Txhua lub kaum sab xis ntawm qhov vertices yog lub kaum sab xis.
• Cov kab pheeb ces kaum sib npaug ntawm qhov ntev, thiab lawv sib cais. Yog li ntawd, cov seem bisected kuj sib npaug ntawm qhov ntev.
• Qhov ntev ntawm kab pheeb ces kaum tuaj yeem suav nrog Pythagoras' theorem:
PQ2 + PS2 =SQ2
• Cov qauv hauv cheeb tsam txo cov khoom ntawm qhov ntev thiab dav.
Area of rectangle=length × width
• Ntau yam khoom sib luag muaj nyob rau ntawm daim duab plaub, xws li;
– Ib daim duab plaub yog voj voog, qhov twg tag nrho cov vertices tuaj yeem muab tso rau ntawm ib puag ncig ntawm lub voj voog.
– Nws yog qhov sib npaug, qhov twg tag nrho cov ces kaum sib npaug.
– Nws yog isogonal, qhov twg txhua lub ces kaum nyob hauv tib lub voj voog.
– Nws muaj ob qho tib si reflectional symmetry thiab rotational symmetry.
Dab tsi yog qhov txawv ntawm Parallelogram thiab Rectangle?
• Parallelogram thiab duab plaub yog quadrilaterals. Daim duab plaub yog qhov tshwj xeeb ntawm cov parallelograms.
• Cheeb tsam ntawm ib qho twg tuaj yeem suav nrog tus qauv hauv qab × qhov siab.
• Xav txog kab pheeb ces kaum;
– Cov kab pheeb ces kaum ntawm qhov parallelogram sib sib zog nqus, thiab sib faib cov kab sib luag los ua ob daim duab peb sab sib dhos.
– Cov kab pheeb ces kaum ntawm cov duab plaub sib npaug ntawm qhov ntev thiab sib npaug; bisected seem yog sib npaug ntawm qhov ntev. Cov kab pheeb ces kaum bisect lub duab plaub rau hauv ob lub congruent txoj cai triangles.
• Xav txog cov kaum sab hauv;
– Tawm sab hauv lub kaum sab xis ntawm lub parallelogram yog sib npaug ntawm qhov loj. Ob lub kaum sab hauv uas nyob ib sab yog ntxiv
– Tag nrho plaub lub kaum sab hauv ntawm daim duab plaub yog kaum sab xis.
• Xav txog ob sab;
– Nyob rau hauv ib qho parallelogram, cov lej ntawm cov squares ntawm ob sab yog sib npaug rau cov lej ntawm cov squares ntawm kab pheeb ces kaum (Parallelogram txoj cai)
– Hauv cov duab plaub, cov lej ntawm cov squares ntawm ob sab uas nyob ib sab yog sib npaug rau cov square ntawm kab pheeb ces kaum ntawm qhov kawg. (Pythagoras' Txoj Cai)
