Inscribed quadrilateral nyob rau hauv ib lub voj voog. Quadrilateral ABCD yog sau nyob rau hauv ib lub voj voog

2024 Tus sau: Angel Austin | [email protected]. Kawg hloov kho: 2024-01-02 17:48

Nrog rau kev faib lej ua lej rau hauv algebra thiab geometry, cov ntaub ntawv kawm nyuaj dua. Cov duab tshiab thiab lawv cov xwm txheej tshwj xeeb tshwm sim. Yuav kom nkag siab txog cov ntaub ntawv zoo, nws yuav tsum tau kawm txog cov ntsiab lus, cov khoom ntawm cov khoom thiab cov theorems muaj feem xyuam.

Cov ntsiab lus dav dav

Ib plaub plaub txhais tau tias daim duab geometric. Nws muaj 4 ntsiab lus. Ntxiv mus, 3 ntawm lawv tsis nyob ntawm tib txoj kab ncaj nraim. Muaj cov ntu txuas cov ntsiab lus tshwj xeeb hauv koob.

Tag nrho cov plaub ntug kawm hauv chav kawm geometry hauv tsev kawm ntawv tau qhia hauv daim duab hauv qab no. Xaus: txhua yam khoom los ntawm daim duab nthuav tawm muaj cov khoom ntawm daim duab dhau los.

A quadrilateral tuaj yeem yog hom hauv qab no:

• Parallelogram. Lub parallelism ntawm nws qhov opposite sab yog ua pov thawj los ntawm cov theorems coj.
• Khawv koob. Ib tug quadrilateral nrog parallel bases. Lwm ob tog tsis yog.
• Rectangle. Ib daim duab uas muaj tag nrho 4 lub ces kaum=90º.
• Rhombus. Daim duab nrog txhua sab sib npaug.
• Square. Muab cov khoom ntawm ob daim duab kawg. Nws muaj txhua sab sib npaug thiab txhua lub kaum sab xis yog.

Lub ntsiab txhais ntawm lub ncauj lus no yog plaub plaub fab sau rau hauv ib lub voj voog. Nws muaj hauv qab no. Qhov no yog ib daim duab nyob ib ncig ntawm lub voj voog tau piav qhia. Nws yuav tsum dhau los ntawm tag nrho cov vertices. Cov kaum sab hauv ntawm plaub plaub fab sau rau hauv ib lub voj voog ntxiv txog 360º.

Tsis yog txhua qhov plaub ntug tuaj yeem sau tau. Qhov no yog vim lub fact tias lub perpendicular bisectors ntawm 4 sab yuav tsis sib tshuam ntawm ib tug taw tes. Qhov no yuav ua rau nws nrhiav tsis tau qhov nruab nrab ntawm lub voj voog ncig 4-gon.

Special case

Muaj kev zam rau txhua txoj cai. Yog li, hauv lub ncauj lus no kuj tseem muaj cov xwm txheej tshwj xeeb:

• Ib qho parallelogram, zoo li, tsis tuaj yeem sau rau hauv ib lub voj voog. Tsuas yog nws rooj plaub tshwj xeeb xwb. Nws yog ib daim duab plaub.
• Yog hais tias tag nrho cov vertices ntawm lub rhombus nyob rau ntawm kab ntawv, ces nws yog square.
• Tag nrho cov vertices ntawm lub trapezoid yog nyob rau ntawm ciam teb ntawm lub voj voog. Hauv qhov no, lawv hais txog daim duab isosceles.

Cov khoom ntawm ib daim ntawv sau plaub npaug hauv lub voj voog

Ua ntej daws cov teeb meem yooj yim thiab nyuaj ntawm ib lub ntsiab lus, koj yuav tsum txheeb xyuas koj qhov kev paub. Yog tsis kawm cov ntaub ntawv kawm, nws tsis tuaj yeem daws tau ib qho piv txwv.

Theorem 1

Cov lej ntawm cov kaum sab xis ntawm lub kaum sab xis sau rau hauv lub voj voog yog 180º.

pov thawj

Muab: quadrilateral ABCD yog sau rau hauv ib lub voj voog. Nws qhov chaw yog taw tes O. Peb yuav tsum ua pov thawj tias _< A + _<C=180º thiab _< B + _<D=180º.

Yuav tsum tau xav txog cov duab qhia.

1. _{<A yog sau rau hauv ib lub voj voog nyob nruab nrab ntawm taw tes O. Nws ntsuas los ntawm ½ BCD (ib nrab arc).}
2. _{<C yog sau rau hauv tib lub voj voog. Nws yog ntsuas los ntawm ½ BAD (ib nrab-arc).}
3. BAD thiab BCD tsim ib lub voj voog tag nrho, piv txwv li lawv qhov loj yog 360º.
4. _<A + _{<C yog sib npaug rau ib nrab ntawm cov lej ntawm ib nrab-arcs sawv cev.}
5. Hence _<A + _{<C=360º / 2=180º.}

Nyob zoo ib yam, cov pov thawj rau _<B thiab _{<D. Txawm li cas los xij, muaj kev daws teeb meem thib ob rau qhov teeb meem.}

1. Nws paub tias qhov sib npaug ntawm cov kaum sab hauv ntawm lub quadrilateral yog 360º.
2. Vim _<A + _<C=180º. Raws li, _<B + _{<D=360º - 180º=180º.}

Theorem 2

(Nws feem ntau hu ua inverse) Yog hais tias nyob rau hauv ib quadrilateral _<A + _<C=180º thiab _< B + _{<D=180º (yog tias lawv tawm tsam), ces ib lub voj voog tuaj yeem piav qhia ib puag ncig xws li daim duab.}

pov thawj

Qhov sib npaug ntawm cov kaum sab xis ntawm plaub fab ABCD sib npaug rau 180º tau muab. _<A + _<C=180º, _<B +_{<D=180º. Peb yuav tsum tau ua pov thawj tias lub voj voog tuaj yeem ncig ncig ABCD.}

Los ntawm chav kawm geometry nws paub tias lub voj voog tuaj yeem kos los ntawm 3 cov ntsiab lus ntawm plaub plaub fab. Piv txwv li, koj tuaj yeem siv cov ntsiab lus A, B, C. Qhov twg yuav taw tes D? Muaj 3 qhov kwv yees:

1. Nws xaus rau hauv lub voj voog. Hauv qhov no, D tsis kov kab.
2. Tawm ntawm lub voj voog. Nws mus deb dhau ntawm kab uas tau teev tseg.
3. Nws tig tawm ntawm lub voj voog.

Nws yuav tsum xav tias D yog nyob rau hauv lub voj voog. Qhov chaw ntawm qhov taw qhia vertex yog nyob ntawm D'. Nws hloov tawm quadrilateral ABCD'.

The result is:_<B + _<D´=2d.

Yog tias peb txuas ntxiv AD' mus rau qhov kev sib tshuam nrog lub voj voog uas twb muaj lawm nyob rau ntawm taw tes E thiab txuas E thiab C, peb tau txais ib daim ntawv sau quadrilateral ABCE. Los ntawm thawj theorem ua raws li kev sib npaug:

Raws li txoj cai ntawm geometry, cov lus qhia tsis siv tau vim _< D' yog lub kaum sab nraud ntawm daim duab peb sab CD'E. Raws li, nws yuav tsum ntau tshaj _{< E. Los ntawm qhov no peb tuaj yeem xaus tias D yuav tsum nyob hauv lub voj voog lossis sab nraud.}

Ib yam li ntawd, qhov kev xav thib peb tuaj yeem ua pov thawj tsis yog thaum D' mus dhau ntawm tus ciam teb ntawm daim duab piav qhia.

Los ntawm ob qhov kev xav ua raws li qhov tseeb xwb. Vertex D yog nyob rau ntawm lub voj voog kab. Hauv lwm lo lus, D coincides nrog E. Nws ua raws li tag nrho cov ntsiab lus ntawm plaub fab nyob rau ntawm kab piav qhia.

Ntawm noob lub theorems, corollaries ua raws:

Txhua daim duab plaub tuaj yeem sau rau hauv ib lub voj voog. Muaj lwm qhov tshwm sim. Ib lub voj voog tuaj yeem ua ib ncig ntawm txhua daim duab plaub

Trapezoid nrog lub duav sib npaug tuaj yeem sau rau hauv lub voj voog. Hauv lwm lo lus, nws suab zoo li no: lub voj voog tuaj yeem piav qhia nyob ib ncig ntawm lub trapezoid nrog cov npoo sib npaug

Ib yam piv txwv

Teeb meem 1. Quadrilateral ABCD yog sau rau hauv ib lub voj voog. _<ABC=105º, _<CAD=35º. Xav nrhiav _{<ABD. Cov lus teb yuav tsum sau ua qib.}

Kev txiav txim siab. Thaum xub thawj, nws yuav zoo li nyuaj nrhiav cov lus teb.

1. Koj yuav tsum nco ntsoov cov khoom ntawm lub ncauj lus no. Namely: cov sum ntawm opposite ces kaum=180º.

_<ADC=180º – _{<ABC=180º – 105º=75º}

Hauv geometry, nws yog qhov zoo dua los ua raws li lub hauv paus ntsiab lus: nrhiav txhua yam koj ua tau. Pab tau tom qab.

2. Cov kauj ruam tom ntej: siv daim duab peb sab sum theorem.

_<ACD=180º – _<CAD – _{<ADC=180º – 35º – 75º=70º}

_<ABD thiab _{<ACD tau sau. Los ntawm cov xwm txheej, lawv cia siab rau ib qho arc. Raws li, lawv muaj qhov sib npaug:}

_<ABD=_<ACD=70º

Answer: _<ABD=70º.

Teeb meem 2. BCDE yog ib daim ntawv sau ua plaub fab hauv ib lub voj voog. _<B=69º, _<C=84º. Qhov chaw ntawm lub voj voog yog taw tes E. Nrhiav - _{< E.}

Kev txiav txim siab.

1. Yuav tsum nrhiav _{<E by Theorem 1.}

_<E=180º – _{<C=180º – 84º=96º}

Answer: _{< E=96º.}

Teeb meem 3. Muab ib daim duab plaub sau ua ib lub voj voog. Cov ntaub ntawv yog qhia nyob rau hauv daim duab. Yuav tsum nrhiav tsis tau qhov tseem ceeb x, y, z.

Kev daws:

z=180º – 93º=87º (los ntawm Theorem 1)

x=½(58º + 106º)=82º

y=180º – 82º=98º (los ntawm Theorem 1)

teb: z=87º, x=82º, y=98º.

Teeb meem 4. Muaj ib lub quadrilateral sau nyob rau hauv ib lub voj voog. Cov txiaj ntsig tau qhia hauv daim duab. Nrhiav x, y.

Kev daws:

x=180º - 80º=100º

y=180º – 71º=109º

Teb: x=100º, y=109º.

Teeb meem rau kev daws teeb meem

Piv txwv 1. Muab ib lub voj voog. Nws qhov chaw yog point O. AC thiab BD yog diameters. _<ACB=38º. Xav nrhiav _{<AOD. Teb yuav tsum muab nyob rau hauv degrees.}

Piv txwv 2. Muab ib lub quadrilateral ABCD thiab ib lub voj voog ncig nws. _<ABC=110º, _<ABD=70º. Nrhiav _{<CAD. Sau koj cov lus teb hauv qib.}

Piv txwv 3. Muab ib lub voj voog thiab ib daim ntawv sau ua plaub fab ABCD. Nws ob lub kaum sab xis yog 82º thiab58º ib. Koj yuav tsum nrhiav qhov loj tshaj plaws ntawm cov ces kaum thiab sau cov lus teb hauv qib.

Piv txwv 4. Quadrilateral ABCD tau muab. Lub kaum sab xis A, B, C tau muab rau hauv qhov sib piv 1: 2: 3. Nws yog ib qho tsim nyog yuav tsum nrhiav lub kaum sab xis D yog tias qhov teev quadrilateral tuaj yeem sau rau hauv lub voj voog. Teb yuav tsum muab nyob rau hauv degrees.

Piv txwv 5. Quadrilateral ABCD muab. Nws ob sab tsim arcs ntawm lub voj voog circumscribed. Degree qhov tseem ceeb AB, BC, CD thiab AD, feem, yog: 78˚, 107˚, 39˚, 136˚. Koj yuav tsum nrhiav _{<Los ntawm lub quadrangle muab thiab sau cov lus teb hauv qib.}