Pythagorean theorem: lub square ntawm lub hypotenuse sib npaug ntawm cov ceg squared

2024 Tus sau: Angel Austin | [email protected]. Kawg hloov kho: 2023-12-17 05:28

Txhua tus menyuam kawm ntawv paub tias qhov square ntawm lub hypotenuse yeej ib txwm sib npaug rau cov lej ntawm ob txhais ceg, txhua qhov yog squared. Cov lus no yog hu ua Pythagorean theorem. Nws yog ib qho ntawm cov theorems nto moo tshaj plaws hauv trigonometry thiab lej feem ntau. Xav paub ntau ntxiv.

Lub tswvyim ntawm daim duab peb sab txoj cai

Ua ntej yuav pib xav txog Pythagorean theorem, nyob rau hauv uas lub square ntawm lub hypotenuse yog sib npaug zos rau cov sum ntawm ob txhais ceg uas yog squared, peb yuav tsum xav txog lub tswvyim thiab cov khoom ntawm lub kaum sab xis ntawm daim duab peb sab, uas lub theorem yog siv tau.

Daim duab peb sab yog daim duab tiaj tus nrog peb lub kaum sab xis thiab peb sab. Ib daim duab peb sab, raws li nws lub npe txhais, muaj ib lub kaum sab xis, uas yog, lub kaum sab xis no yog 90o.

Los ntawm cov khoom dav dav rau txhua daim duab peb sab, nws paub tias cov lej ntawm tag nrho peb lub kaum ntawm daim duab no yog 180^o, uas txhais tau hais tias rau daim duab peb sab txoj cai tus lej ntawm ob lub ces kaum uas tsis yog, yog 180^o -90^o=90^o. Qhov tseeb kawg txhais tau tias txhua lub kaum sab xis ntawm daim duab peb sab uas tsis yog lub kaum sab xis yuav ib txwm tsawg dua 90^o.

Ib sab uas nyob sab nraud ntawm lub kaum sab xis yog hu ua hypotenuse. Lwm ob sab yog ob txhais ceg ntawm daim duab peb sab, lawv tuaj yeem sib npaug rau ib leeg, lossis lawv tuaj yeem sib txawv. Nws paub los ntawm trigonometry tias qhov ntau dua lub kaum sab xis uas ib sab nyob hauv daim duab peb sab, qhov ntev ntawm sab no ntau dua. Qhov no txhais tau hais tias nyob rau hauv ib tug txoj cai daim duab peb sab lub hypotenuse (pw opposite lub kaum sab xis 90 ^o) yeej ib txwm yuav loj dua ib tug ntawm ob txhais ceg (dag opposite lub ces kaum < 90 ^o).

lej cim ntawm Pythagorean theorem

Qhov kev xav no hais tias lub xwmfab ntawm lub hypotenuse yog sib npaug rau cov lej ntawm ob txhais ceg, txhua qhov uas yav dhau los yog squared. Txhawm rau sau qhov kev tsim lej no, xav txog ib daim duab peb sab uas sab a, b, thiab c yog ob txhais ceg thiab hypotenuse, feem. Nyob rau hauv cov ntaub ntawv no, lub theorem, uas yog teev raws li lub square ntawm lub hypotenuse yog sib npaug zos rau cov sum ntawm cov squares ntawm ob txhais ceg, tuaj yeem sawv cev los ntawm cov qauv hauv qab no: c²=a. 2 ^{+ b} 2^{. Los ntawm no, lwm cov qauv tseem ceeb rau kev xyaum tuaj yeem tau txais: a=√(c}2 ^{- b}2^{), b=√(c} 2 ^{- a}2^{) and c=√(a}2 ^{+ b}2^).

Nco ntsoov tias nyob rau hauv rooj plaub ntawm txoj cai-angled equilateral daim duab peb sab, uas yog, a=b, lub formulation: lub square ntawm lub hypotenuse yog sib npaug zos rau cov sum ntawm ob txhais ceg, txhua yam uassquared, lej sau li: c²=a² + b²=2a ^{2, uas txhais tau tias kev sib npaug: c=a√2.}

keeb kwm keeb kwm

Pythagorean theorem, uas hais tias lub xwmfab ntawm lub hypotenuse yog sib npaug rau cov lej ntawm ob txhais ceg, txhua tus uas yog squared, tau paub ntev ua ntej tus naas ej Greek philosopher tau mloog nws. Ntau cov papyri ntawm Egypt thaum ub, nrog rau cov ntsiav tshuaj av nplaum ntawm Babylonians, paub meej tias cov neeg no tau siv cov cuab yeej sau tseg ntawm ob sab ntawm daim duab peb sab. Piv txwv li, ib qho ntawm thawj Iyiv pyramids, lub Pyramid ntawm Khafre, uas nws kev tsim kho hnub rov qab mus rau 26th caug xyoo BC (2000 xyoo ua ntej lub neej ntawm Pythagoras), tau tsim los ntawm kev paub txog qhov piv ntawm 3x4x5 txoj cai daim duab peb sab.

Vim li cas tam sim no lub theorem muaj npe tom qab Greek? Cov lus teb yog yooj yim: Pythagoras yog thawj zaug ua lej ua pov thawj no theorem. Ciaj sia nyob Babylonian thiab Egyptian cov ntawv sau tsuas yog hais txog nws txoj kev siv, tab sis tsis muab cov pov thawj lej.

Nws ntseeg tias Pythagoras tau ua pov thawj lub theorem nyob rau hauv kev txiav txim siab los ntawm kev siv cov khoom ntawm cov duab peb sab zoo sib xws, uas nws tau txais los ntawm kev kos ib qhov siab ntawm daim duab peb sab ntawm lub kaum sab xis 90o rau lub hypotenuse.

Ib qho piv txwv ntawm kev siv Pythagorean theorem

Xav txog qhov teeb meem yooj yim: nws yog ib qho tsim nyog los txiav txim siab qhov ntev ntawm qhov inclined staircase L, yog tias nws paub tias nws muaj qhov siab H=3meters, thiab qhov deb ntawm phab ntsa tiv thaiv uas tus ntaiv so rau nws ko taw yog P=2.5 meters.

Hauv qhov no, H thiab P yog ob txhais ceg, thiab L yog hypotenuse. Txij li qhov ntev ntawm lub hypotenuse yog sib npaug rau cov lej ntawm cov squares ntawm ob txhais ceg, peb tau txais: L²=H² + P 2^{, whence L=√(H}2 ^{+ P}2^)=√(3 2 ^{+ 2, 5} 2^{)=3.905 meters lossis 3 meters thiab 90.5 cm.}