Ib daim duab peb sab yog polygon nrog peb sab (peb fab). Feem ntau, ob sab yog qhia los ntawm cov ntawv me, sib xws rau cov ntawv loj uas qhia txog cov vertices opposite. Nyob rau hauv tsab xov xwm no, peb yuav tau paub txog cov hom ntawm cov duab geometric, lub theorem uas txiav txim siab seb qhov sum ntawm lub kaum sab xis ntawm daim duab peb sab yog dab tsi.
Views by angles
Cov nram qab no hom polygon nrog peb vertices yog qhov txawv:
- mob-lub kaum sab xis, hauv txhua lub ces kaum ntse;
- daim duab plaub, muaj ib lub kaum sab xis, thaum ob sab uas tsim nws hu ua ob txhais ceg, thiab sab uas muab tso rau sab xis ntawm lub kaum sab xis yog hu ua hypotenuse;
- obtuse thaum ib ces kaum yog obtuse;
- isosceles, uas ob sab sib npaug, thiab lawv hu ua lateral, thiab qhov thib peb yog lub hauv paus ntawm daim duab peb sab;
- equilateral, muaj tag nrho peb sab sib npaug.
Properties
Lawv qhia txog cov khoom tseem ceeb uas yog yam ntxwv ntawm txhua hom duab peb sab:
- pua sab loj dua ib txwm muaj lub kaum ntse dua, thiab rov ua dua;
- sab nraud ntawm qhov sib npaug yog cov kaum sib npaug, thiab rov ua dua;
- ib daim duab peb sab muaj ob lub kaum ntse ntse;
- ib lub kaum sab nraud loj dua ib lub kaum sab hauv tsis nyob ib sab;
- qhov sum ntawm ob lub ces kaum ib txwm tsawg dua 180 degrees;
- kaum sab nrauv sib npaug ntawm ob lub ces kaum uas tsis cuam tshuam nrog nws.
Daim duab peb sab ntawm lub kaum sab xis
Lub theorem hais tias yog tias koj ntxiv tag nrho cov kaum sab xis ntawm daim duab geometric, uas nyob ntawm lub dav hlau Euclidean, ces lawv cov lej yuav yog 180 degrees. Cia peb sim ua pov thawj qhov theorem.
Cia peb muaj arbitrary daim duab peb sab nrog vertices ntawm KMN.
Los ntawm vertex M kos ib txoj kab ncaj nraim mus rau txoj kab ncaj nraim KN (txoj kab no tseem hu ua Euclidean ncaj kab). Peb khij lub ntsiab lus A ntawm nws txoj hauv kev uas cov ntsiab lus K thiab A nyob ntawm ntau sab ntawm txoj kab ncaj nraim MN. Peb tau txais cov ces kaum sib npaug AMN thiab KNM, uas, zoo li cov sab hauv, dag crosswise thiab tsim los ntawm secant MN ua ke nrog cov kab ncaj KN thiab MA, uas yog sib npaug. Los ntawm qhov no nws ua raws li cov sum ntawm cov kaum sab xis ntawm daim duab peb sab nyob rau ntawm lub vertices M thiab H yog sib npaug rau qhov loj ntawm lub kaum sab xis KMA. Tag nrho peb lub kaum ntse ntse tsim cov lej, uas yog sib npaug rau cov lej ntawm cov ces kaum KMA thiab MKN. Vim cov ces kaum yog sab hauv ib sab nrog hwmCov kab ncaj nraim KN thiab MA nrog lub secant KM, lawv cov lej yog 180 degrees. Theorem proven.
Txhob Cia Siab
Cov kab lus hauv qab no ua raws li qhov kev xav tau ua pov thawj saum toj no: ib daim duab peb sab muaj ob lub kaum ntse ntse. Txhawm rau ua pov thawj qhov no, cia peb xav tias daim duab geometric muab tsuas muaj ib lub kaum sab xis. Nws tseem tuaj yeem xav tias tsis muaj lub kaum ntse ntse. Hauv qhov no, yuav tsum muaj tsawg kawg yog ob lub kaum sab xis uas sib npaug lossis siab dua 90 degrees. Tab sis tom qab ntawd tus lej ntawm cov ces kaum yuav ntau dua 180 degrees. Tab sis qhov no tsis tuaj yeem ua tau, vim hais tias raws li lub theorem, cov lej ntawm lub kaum sab xis ntawm daim duab peb sab yog 180 ° - tsis muaj ntau thiab tsis tsawg. Nov yog qhov yuav tsum tau ua pov thawj.
Sab nrauv ces kaum khoom
Dab tsi yog cov lej ntawm cov kaum ntawm daim duab peb sab uas yog sab nraud? Lo lus nug no tuaj yeem teb tau hauv ib qho ntawm ob txoj hauv kev. Thawj qhov yog tias nws yog qhov yuav tsum tau nrhiav cov lej ntawm cov ces kaum, uas tau coj ib qho ntawm txhua qhov vertex, uas yog, peb lub kaum sab xis. Qhov thib ob txhais tau hais tias koj yuav tsum nrhiav cov lej ntawm tag nrho rau lub kaum sab xis ntawm lub vertices. Ua ntej, cia peb tham nrog thawj qhov kev xaiv. Yog li, daim duab peb sab muaj rau sab nraud - ob ntawm txhua qhov vertex.
Txhua khub muaj kaum sib npaug vim tias lawv yog ntsug:
∟1=∟4, ∟2=∟5, ∟3=∟6.
Dhau li ntawm, nws paub tias lub kaum sab nraud ntawm daim duab peb sab yog sib npaug rau cov lej ntawm ob lub kaum sab hauv uas tsis cuam tshuam nrog nws. Yog li ntawd, ∟1=∟A + ∟C, ∟2=∟A + ∟B, ∟3=∟B + ∟C.
Los ntawm qhov no nws hloov tawm tias cov lej ntawm sab nraudcov ces kaum, uas muab ib qho ntawm txhua qhov vertex, yuav muab sib npaug rau:
∟1 + ∟2 + ∟3=∟A + ∟C + ∟A + ∟B + ∟B + ∟C=2 x (∟A + ∟B + ∟C).
Tau tias qhov sib npaug ntawm cov kaum sab xis yog 180 °, nws tuaj yeem sib cav tias ∟A + ∟B + ∟C=180 °. Thiab qhov no txhais tau tias ∟1 + ∟2 + ∟3=2 x 180°=360°. Yog tias qhov kev xaiv thib ob yog siv, ces tus lej ntawm rau lub kaum sab xis yuav yog, feem, ob zaug ntau dua. Ntawd yog, cov lej ntawm cov kaum sab nraud ntawm daim duab peb sab yuav yog:
∟1 + ∟2 + ∟3 + ∟4 + ∟5 + ∟6=2 x (∟1 + ∟2 + ∟2)=720°.
daim duab peb sab
Dab tsi yog cov lej ntawm cov kaum sab xis ntawm daim duab peb sab? Cov lus teb rau lo lus nug no, dua, ua raws li lub theorem, uas hais tias cov ces kaum nyob rau hauv ib daim duab peb sab ntxiv mus txog 180 degrees. Thiab peb cov lus (cov khoom vaj khoom tsev) suab zoo li no: nyob rau hauv txoj cai daim duab peb sab, cov kaum sab xis ntxiv mus txog 90 degrees. Cia peb ua pov thawj nws qhov tseeb.
Cia peb muab ib daim duab peb sab KMN, uas ∟Н=90°. Yuav tsum ua pov thawj tias ∟K + ∟M=90°.
Yog li, raws li lub kaum ntse ntse theorem ∟К + ∟М + ∟Н=180 °. Peb qhov xwm txheej hais tias ∟Н=90 °. Yog li nws hloov tawm, ∟K + ∟M + 90 °=180 °. Ntawd yog, ∟K + ∟M=180° - 90°=90°. Nov yog qhov peb yuav tsum tau ua pov thawj.
Ntxiv rau cov khoom saum toj no ntawm daim duab peb sab, koj tuaj yeem ntxiv cov hauv qab no:
- kaum ob txhais ceg yog ntse;
- lub hypotenuse yog daim duab peb sab ntau dua li ob txhais ceg;
- tus lej ntawm ob txhais ceg ntau dua qhov hypotenuse;
- legib daim duab peb sab uas nyob ib sab ntawm lub kaum sab xis ntawm 30 degrees yog ib nrab ntawm lub hypotenuse, uas yog, sib npaug rau ib nrab ntawm nws.
Raws li lwm yam khoom ntawm daim duab geometric no, Pythagorean theorem tuaj yeem paub qhov txawv. Nws hais tias nyob rau hauv ib daim duab peb sab nrog ib tug kaum sab xis ntawm 90 degrees (pob tshab), cov sum ntawm cov squares ntawm ob txhais ceg yog sib npaug zos rau lub square ntawm lub hypotenuse.
Cov lej ntawm cov kaum ntawm daim duab peb sab isosceles
Ua ntej peb tau hais tias isosceles yog polygon nrog peb lub vertices, muaj ob sab sib npaug. Cov cuab yeej ntawm ib daim duab geometric yog paub: cov ces kaum ntawm nws lub hauv paus yog sib npaug. Cia peb ua pov thawj.
Tau daim duab peb sab KMN, uas yog isosceles, KN yog nws lub hauv paus.
Peb yuav tsum ua pov thawj tias ∟К=∟Н. Yog li, cia peb hais tias MA yog lub bisector ntawm peb daim duab peb sab KMN. Lub MCA daim duab peb sab, coj mus rau hauv tus account thawj kos npe ntawm kev sib npaug, yog sib npaug rau MCA daim duab peb sab. Namely, los ntawm cov xwm txheej nws tau muab tias KM=NM, MA yog ib sab, ∟1=∟2, vim MA yog bisector. Siv qhov tseeb tias ob daim duab peb sab sib npaug, peb tuaj yeem hais tias ∟K=∟Н. Yog li qhov theorem tau ua pov thawj.
Tab sis peb txaus siab rau qhov sum ntawm lub kaum sab xis ntawm daim duab peb sab (isosceles). Txij li thaum nyob rau hauv no hwm nws tsis muaj nws tus kheej peculiarities, peb yuav pib los ntawm lub theorem xam ua ntej. Qhov ntawd yog, peb tuaj yeem hais tias ∟K + ∟M + ∟H=180°, lossis 2 x ∟K + ∟M=180° (txij li ∟K=∟H). Peb yuav tsis ua pov thawj cov cuab yeej no, txij li daim duab peb sab sum theorem nws tus kheej tau ua pov thawj ua ntej.
Tsuas yog raws li tau thamCov khoom hais txog lub kaum sab xis ntawm daim duab peb sab, kuj tseem muaj cov lus tseem ceeb xws li:
- nyob rau hauv daim duab peb sab isosceles, qhov siab uas tau txo qis rau hauv lub hauv paus yog ob qho tib si nruab nrab, lub bisector ntawm lub kaum sab xis ntawm qhov sib npaug, nrog rau cov axis ntawm symmetry ntawm nws lub hauv paus;
- medians (bisectors, heights) uas yog kos rau ob sab ntawm cov duab geometric yog sib npaug.
Daim duab peb sab sib npaug
Nws tseem hu ua txoj cai, nws yog daim duab peb sab nrog txhua sab sib npaug. Yog li ntawd, cov ces kaum kuj sib npaug. Txhua tus yog 60 degrees. Cia peb ua pov thawj qhov khoom no.
xav tias peb muaj daim duab peb sab KMN. Peb paub tias KM=NM=KN. Thiab qhov no txhais tau hais tias raws li cov cuab yeej ntawm lub kaum sab xis nyob ntawm lub hauv paus hauv daim duab peb sab isosceles, ∟К=∟М=∟Н. Raws li cov theorem, cov sum ntawm lub kaum sab xis ntawm ib daim duab peb sab yog ∟К + ∟М + ∟Н=180°, ces 3 x ∟К=180° los ∟К=60°, ∟М=60°, ∟ N=60°. Yog li, nqe lus tau ua pov thawj.
Raws li koj tuaj yeem pom los ntawm cov pov thawj saum toj no raws li lub theorem, qhov sib npaug ntawm cov kaum sab xis ntawm daim duab peb sab sib npaug, zoo li cov lej ntawm cov kaum ntawm lwm daim duab peb sab, yog 180 degrees. Tsis tas yuav ua pov thawj qhov theorem dua.
Tseem muaj cov yam ntxwv zoo li ntawm daim duab peb sab sib npaug:
- nruab nrab, bisector, qhov siab hauv cov duab geometric yog tib yam, thiab lawv qhov ntev yog xam raws li (a x √3): 2;
- yog tias koj piav qhia lub voj voog ncig ib puag ncig ib puag ncig, ces nws lub vojvoog yuav yogsib npaug (a x √3): 3;
- yog tias koj sau lub voj voog rau hauv daim duab peb sab sib npaug, ces nws lub vojvoog yuav yog (a x √3): 6;
- cheeb tsam ntawm daim duab geometric yog xam los ntawm tus qauv: (a2 x √3): 4.
Obt-angled daim duab peb sab
Raws li lub ntsiab txhais ntawm daim duab peb sab obtuse, ib qho ntawm nws cov ces kaum yog nyob nruab nrab ntawm 90 thiab 180 degrees. Tab sis muab hais tias ob lub kaum sab xis ntawm daim duab geometric no yog qhov mob hnyav, peb tuaj yeem xaus tias lawv tsis tshaj 90 degrees. Yog li, daim duab peb sab sum ntawm lub kaum sab xis theorem ua haujlwm thaum xam cov sum ntawm cov ces kaum hauv daim duab peb sab obtuse. Nws hloov tawm tias peb tuaj yeem hais yam xyuam xim, raws li cov lus hais saum toj no, tias cov lej ntawm cov kaum sab xis ntawm daim duab peb sab obtuse yog 180 degrees. Ntxiv dua thiab, qhov kev xav no tsis tas yuav rov ua pov thawj.
