Lub sum ntawm cov ces kaum ntawm ib tug daim duab peb sab. Lub theorem rau lub sum ntawm cov ces kaum ntawm ib tug daim duab peb sab

Daim duab peb sab yog ib tug polygon muaj peb sab (peb cov ces kaum). Feem ntau cov feem ntau, qhov denoted los ntawm me me tsiaj ntawv coj capital cov tsiaj ntawv, uas sawv cev rau opposite vertices. Nyob rau hauv no tsab xov xwm peb muab ib tug saib nyob rau hauv cov hom ntawm geometric duab, theorem, uas txhais tau tias dab tsi yog sib npaug zos rau qhov tawm ntawm cov ces kaum ntawm ib tug daim duab peb sab.

Hom coob ces kaum

Cov nram qab no hom ntawm polygon nrog peb vertices:

• mob-angled, nyob rau hauv uas tag nrho cov ces kaum ntse;
• lub muaj ib txoj cai lub, lub sab txoj kev ua nws, xa mus rau ob txhais ceg, thiab sab uas yog pov tseg opposite mus rau sab xis lub kaum sab xis yog hu ua tus hypotenuse;
• obtuse thaum muaj ib tug lub yog obtuse ;
• ceskaum, uas nws ob sab yog muaj sib npaug, thiab lawv yog hu ua sab, thiab lub thib peb - ib tug peb tog nrog ib lub hauv paus;
• equilateral muaj peb sib npaug zos sab.

zog

Faib cov yooj yim zog uas yog cov yam ntxwv ntawm txhua hom ntawm daim duab peb sab:

• rov qab lub qhov loj tshaj ces yog yeej ib txwm ntau dua lub kaum ntse ntse, thiab lwm versa;
• yog muaj sib npaug ces kaum opposite lub sib npaug zos-loj tshaj plaws tog, thiab vice versa;
• nyob rau hauv tej daim duab peb sab muaj ob tug mob ces kaum;
• txheej lub siab tshaj tej internal lub tsis nyob ib sab thereto;
• lub sum ntawm ob ces kaum yog ib txwm tsawg tshaj li 180 degrees;
• sab nrauv lub sib npaug rau hauv lub sum ntawm cov lwm yam ob ces kaum, uas yog tsis mezhuyut nrog nws.

Lub theorem rau lub sum ntawm cov ces kaum ntawm ib tug daim duab peb sab

Lub theorem hais tias yog koj ntxiv li tag nrho cov ces kaum ntawm lub geometric zoo, uas yog nyob rau hauv lub Euclidean dav hlau, ces lawv sum yuav 180 degrees. Wb sim los ua pov thawj no theorem.

Cia peb muaj ib tug arbitrary daim duab peb sab nrog vertices KMN. Nyob rau sab saum toj ntawm M yuav tuav ib tug ncaj qha mus tib seem rau txoj kab KN (txawm no kab yog hu ua Euclid). Nws yuav tsum tau muab sau tseg point A yog li ntawd lub ntsiab lus K thiab A cov txheej txheem los ntawm txawv sab ntawm txoj kab MN. Peb tau txais tib yam kaum sab xis ntawm AMS thiab MUF, uas, xws li rau sab hauv, pw crosswise mus rau daim ntawv kev sib tshuam MN nyob rau hauv nrog direct CN thiab MA, uas yog mus tib seem. Los ntawm no nws hais ntxiv hais tias lub sum ntawm cov ces kaum ntawm daim duab peb sab, nyob rau ntawm lub vertices ntawm M thiab N yog sib npaug zos rau qhov luaj li cas ntawm cov CMA kaum sab xis. Tag nrho peb cov ces kaum muaj ib tug sum sib npaug zos rau qhov tawm ntawm cov ces kaum ntawm KMA thiab MCS. Txij li thaum cov ntaub ntawv yog cov ces kaum txheeb ze mog thaum uas tig mus kab CL thiab CM MA ntawm kev sib tshuam, lawv zaum yog 180 degrees. Qhov no qhia tau cov theorem.

tshwm sim

Ntawm saum toj no lub saum toj no theorem implies lub nram qab no corollary: txhua txhua daim duab peb sab muaj ob tug mob ces kaum. Los ua pov thawj no, qhia rau peb xav tias hais tias qhov no geometrical daim duab muaj tsuas yog ib mob kaum sab xis. Koj yuav tau xav hais tias tsis muaj ntawm cov fab yog tsis ntse. Nyob rau hauv cov ntaub ntawv no nws yuav tsum muaj tsawg kawg yog ob lub ces kaum, qhov ntau ntawm cov uas yog sib npaug los yog ntau dua 90 degrees. Tab sis, ces lub sum ntawm cov ces kaum ntau tshaj 180 degrees. Tab sis qhov no tsis tau yuav, raws li raws li lub theorem sum ces kaum ntawm ib tug peb tog yog sib npaug zos rau 180 ° - tsis muaj ntau, tsis muaj tsawg. Hais tias yog dab tsi yuav tsum tau muab pov thawj.

Khoom tawm sab nraum ntawm cov ces kaum

Yuav ua li cas yog lub sum ntawm cov ces kaum ntawm ib tug daim duab peb sab, uas yog sab nraud? Lo lus teb rau lo lus nug no yuav muab tau los ntawm kev thov ib yam ntawm ob txoj kev. Tus thawj yog hais tias koj yuav tau mus nrhiav lub sum ntawm cov ces kaum, uas yog npaum li cas ib tug nyob rau txhua kab los sib ntsib, uas yog, peb cov ces kaum. Qhov thib ob implies hais tias koj yuav tau mus nrhiav lub sum ntawm rau ces kaum ntawm lub vertices. Yuav kom nrog qhov pib ntawm thawj embodiment. Yog li, daim duab peb sab muaj rau txheej fab - nyob rau ntawm sab saum toj ntawm txhua tus ntawm ob tug. Txhua khub muaj sib npaug zos ces kaum ntawm lawv tus kheej, txij thaum lawv yog ntsug:

∟1 = ∟4, ∟2 = ∟5, ∟3 = ∟6.

Nyob rau hauv tas li ntawd, nws yog lub npe hu hais tias tus txheej ces kaum ntawm ib tug daim duab peb sab sib npaug rau hauv lub sum ntawm ob sab hauv, uas yog tsis mezhuyutsya nrog nws. yog li ntawd,

∟1 = ∟A + ∟S, ∟2 = ∟A + ∟V, ∟3 = ∟V + ∟S.

Los ntawm no nws zoo nkaus li uas lub sum ntawm cov sab nrauv ces kaum, uas yog npaum li cas ib tug los ntawm ib tug nyob ze txhua kab los sib ntsib yuav sib npaug zos rau:

∟1 + ∟2 + ∟3 = ∟A + + ∟S ∟A ∟V + + + ∟V ∟S = 2 x (∟A + ∟V ∟S +).

Muab qhov tseeb hais tias lub sum ntawm cov ces kaum sib npaug 180 degrees, nws yuav sib cav hais tias ∟A + ∟V ∟S = + 180 °. Qhov no txhais tau tias ∟1 + ∟2 + ∟3 = 2 x 180 ° = 360 °. Yog hais tias lub kev xaiv thib ob yog siv, tus lej rau cov ces kaum yuav correspondingly ntau dua ob zaug. Piv txwv li lub sum ntawm cov ces kaum ntawm ib tug peb tog sab nraum yuav tsum:

∟1 + ∟2 + ∟3 + ∟4 + ∟5 + ∟6 = 2 x (∟1 + ∟2 + ∟2) = 720 °.

txoj cai daim duab peb sab

Yuav ua li cas yog sib npaug zos rau cov sum ntawm cov ces kaum ntawm ib txoj cai daim duab peb sab, yog cov kob? Cov lus teb no, dua, los ntawm Theorem, uas hais tias cov ces kaum ntawm ib tug daim duab peb sab ntxiv mus txog 180 degrees. A suab peb assertion (vaj tse) raws li nram no: nyob rau hauv ib txoj cai daim duab peb sab ntse ces kaum ntxiv mus txog 90 degrees. Peb ua pov thawj nws veracity. Cia muaj muab daim duab peb sab KMN, uas ∟N = 90 °. Nws yog tsim nyog los ua pov thawj hais tias ∟K ∟M = + 90 °.

Yog li, raws li lub theorem rau lub sum ntawm cov ces kaum ∟K + ∟M ∟N + = 180 °. Nyob rau hauv no mob nws yog hais tias ∟N = 90 °. Nws puv tawm ∟K ∟M + + 90 ° = 180 °. Qhov ntawd yog ∟K ∟M + = 180 ° - 90 ° = 90 °. Hais tias yog dab tsi peb yuav tsum los ua pov thawj.

Nyob rau hauv tas li ntawd mus rau saum toj thaj chaw ntawm ib txoj cai daim duab peb sab, koj muaj peev xwm ntxiv no:

• ces kaum, uas pw tiv thaiv ob txhais ceg yog ntse;
• lub hypotenuse ntawm lub voos loj dua ib yam ntawm cov ob txhais ceg;
• lub sum ntawm ob txhais ceg ntau dua qhov hypotenuse;
• ceg ntawm daim duab peb sab, uas lies opposite rau lub kaum sab xis ntawm 30 degrees, ib nrab ntawm cov hypotenuse, uas yog sib npaug zos rau nws ib nrab.

Raws li ib cuab yeej ntawm tus duab zoo yuav txawv Pythagorean theorem. Nws cav hais tias nyob rau hauv ib tug daim duab peb sab uas muaj ib qho kaum sab xis ntawm 90 degrees (lub), lub sum ntawm cov plaub fab ntawm ob txhais ceg sib npaug rau hauv lub square ntawm lub hypotenuse.

Lub sum ntawm cov ces kaum ntawm ib daim duab peb sab ceskaum

Nyuam qhuav pib peb hais tias ib tug ceskaum daim duab peb sab yog ib tug polygon nrog peb vertices, muaj ob sib npaug zos sab. Qhov no tej khoom vaj tse yog lub npe hu geometrical daim duab: lub ces kaum ntawm nws lub hauv paus sib npaug zos. Cia peb ua pov thawj no.

Siv daim duab peb sab KMN, uas yog ceskaum, SC - nws puag. Peb yuav tsum ua pov thawj tias ∟K = ∟N. Yog li ntawd, cia peb xav hais tias MA - KMN yog lub bisector ntawm peb daim duab peb sab. ICA daim duab peb sab nrog tus thawj kos npe rau ntawm koob pheej ntawm lawv yog daim duab peb sab mnÄ. Namely, los ntawm kev xav muab hais tias CM = NM, MA yog ib qho sab, ∟1 = ∟2, vim hais tias MA - qhov no bisector. Siv cov koob pheej ntawm lawv ntawm ob cov vajvoos peb sab, ib tug yuav cam hais tias ∟K = ∟N. Li no, lub theorem yog muaj pov thawj.

Tab sis peb xav nyob rau hauv, dab tsi yog qhov tawm ntawm cov ces kaum ntawm ib tug daim duab peb sab (ceskaum). Vim hais tias nyob rau hauv no hwm nws tsis muaj nws cov nta, peb yuav pib los ntawm lub theorem sib tham yav tas los. Hais tias yog, peb yuav hais tias ∟K + ∟M ∟N + = 180 °, los yog 2 x ∟K ∟M + = 180 ° (raws li ∟K = ∟N). Qhov no yuav tsis ua pov thawj cov cuab yeej, raws li tus theorem rau lub sum ntawm cov ces kaum ntawm ib tug peb tog twb muaj pov thawj ua ntej lawm.

Tshwj qhov suav hais tias thaj chaw ntawm lub ces kaum ntawm ib tug daim duab peb sab, muaj kuj yog xws li ib qho tseem ceeb cov lus:

• nyob rau hauv ib tug equilateral triangle qhov siab, uas tau lowered mus rau hauv lub hauv paus, yog ib txhij theem nrab bisector ntawm lub kaum sab xis uas yog nruab nrab ntawm lub sib npaug zos sab thiab lub axis ntawm symmetry ntawm nws puag;
• median (bisector, altitude), uas yog muaj nyob rau mus rau tej sab ntawm ib tug geometric duab, yog muaj sib npaug.

equilateral triangle

Nws tseem hu ua txoj cai, yog daim duab peb sab, uas yog muaj sib npaug rau txhua tog. Thiab yog li ntawd kuj sib npaug thiab cov ces kaum. Txhua tus ntawm lawv yog 60 degrees. Cia peb ua pov thawj cov cuab yeej no.

Cia peb xav hais tias peb muaj ib tug daim duab peb sab KMN. Peb paub tias KM = HM = KH. Qhov no txhais tau tias, raws li cov khoom teejtug uas yog cov ces kaum nyob rau hauv lub puag nyob rau hauv ib tug equilateral triangle ∟K = ∟M = ∟N. Txij li thaum, raws li lub sum ntawm cov ces kaum ntawm ib tug daim duab peb sab theorem ∟K + ∟M ∟N + = 180 °, ces x 3 = 180 ° ∟K los yog ∟K = 60 °, ∟M = 60 °, ∟N = 60 °. Yog li, lub assertion yog muaj pov thawj. Raws li pom los ntawm cov saum toj no pov thawj raws li nyob rau hauv lub saum toj no theorem, lub sum ntawm cov ces kaum ntawm ib daim duab peb sab equilateral, raws li lub sum ntawm cov ces kaum uas muaj lwm yam daim duab peb sab yog 180 degrees. Dua tseeb tiag no theorem yog tsis tsim nyog.

Muaj tseem ib co zog yam ntxwv ntawm ib tug equilateral triangle:

• median bisector siab nyob rau hauv ib tug geometrical daim duab zoo tib yam, thiab lawv ntev yog xam raws li (ib tug x √3): 2;
• Yog hais tias qhov no polygon circumscribing lub vajvoog, ces tus voos kheej-kheej yuav ua ncaj rau (a x √3): 3;
• yog inscribed nyob rau hauv ib lub voj voog equilateral daim duab peb sab, nws voos kheej-kheej yuav (a x √3): 6;
• cheeb tsam ntawm lub geometric duab yog xam los ntawm cov mis: (a2 x √3): 4.

obtuse daim duab peb sab

Los ntawm txhais, ib tug obtuse-angled daim duab peb sab, ib qho ntawm nws cov ces kaum yog nruab nrab ntawm 90 mus rau 180 degrees. Tab sis muab qhov tseeb hais tias cov lwm yam ob ces kaum ntawm lub geometric zoo ntse, nws muaj peev xwm yuav xaus lus tias lawv tsis pub tshaj 90 degrees. Yog li ntawd, lub sum ntawm cov ces kaum ntawm ib tug daim duab peb sab theorem ua hauj lwm nyob rau hauv kev xam xyuas cov sum ntawm cov ces kaum nyob rau hauv ib tug obtuse daim duab peb sab. Yog li ntawd, peb yuav xyuam xim hais tias, raws li nyob rau hauv lub saum toj no theorem hais tias tus sum ntawm cov obtuse kaum ntawm ib tug peb tog yog 180 degrees. Ib zaug ntxiv, qhov no theorem tsis tau rov-ntawv pov thawj.