Pythagorean theorem piav qhia qhov ntev ntawm ob sab ntawm daim duab peb sab txoj hauv kev zoo nkauj thiab siv tau, yog li cov theorem no tseem siv dav rau niaj hnub no. Cov theorem no hais tias rau txhua txoj cai daim duab peb sab, qhov sib npaug ntawm cov plaub fab ntawm ob sab uas tsis yog lub kaum sab xis yog sib npaug rau cov xwm txheej ntawm hypotenuse. Hauv lwm lo lus, rau txoj cai daim duab peb sab nrog ib sab sib npaug a thiab b thiab hypotenuse c, a2 + b2 = c2.
Pythagorean theorem yog ib qho ntawm cov hauv paus ntsiab lus ntawm qib pib geometry. Muaj ntau qhov kev thov suav nrog siv cov qauv no, piv txwv li, kom yooj yim nrhiav qhov nrug ntawm ob lub ntsiab lus ntawm lub dav hlau sib koom ua ke.
Kauj ruam
Txoj Kev 1 ntawm 2: Nrhiav Ib Sab Ntawm Ib Daim Duab Peb Hlis
Kauj Ruam 1. Nco ntsoov tias koj daim duab peb sab yog daim duab peb sab xis
Pythagorean theorem tsuas yog siv rau daim duab peb sab peb sab xis, yog li, ua ntej yuav pib, nws yog ib qho tseem ceeb heev kom ntseeg tau tias koj daim duab peb sab ua tau raws li cov yam ntxwv ntawm daim duab peb sab. Hmoov zoo, muaj ib yam uas tuaj yeem qhia tau tias koj daim duab peb sab yog daim duab peb sab xis. Koj daim duab peb sab yuav tsum muaj ib lub 90 degree.
Raws li lub cim, txoj cai daim duab peb sab feem ntau yog cim nrog cov duab me me los cim 90-degree cov ces kaum, tsis txhob siv cov "nkhaus" nkhaus. Nrhiav qhov cim tshwj xeeb no nyob rau ces kaum ntawm koj daim duab peb sab
Kauj Ruam 2. Muab qhov hloov pauv a, b, thiab c rau ob sab ntawm koj daim duab peb sab
Hauv Pythagorean Theorem, qhov sib txawv a thiab b sawv cev rau ob sab uas sib ntsib ntawm daim duab peb sab sab xis, thaum qhov sib txawv c sawv cev rau qhov hypotenuse - sab ntev sib txawv ntawm lub kaum sab xis. Yog li, txhawm rau pib nrog, kos rau sab luv ntawm koj daim duab peb sab nrog cov hloov pauv a thiab b (nws tsis muaj teeb meem yog tias koj pauv lawv), thiab khij qhov hypotenuse nrog qhov sib txawv c.
Kauj Ruam 3. Txiav txim sab twg ntawm daim duab peb sab uas koj xav daws
Pythagorean theorem tso cai rau cov lej lej nrhiav qhov ntev ntawm ib sab ntawm daim duab peb sab txoj cai tsuav lawv paub qhov ntev ntawm ob sab. Txiav txim sab twg tsis paub - a, b, thiab/lossis c. Yog tias qhov ntev ntawm ib sab ntawm koj sab tsis paub, koj npaj txhij txav mus.
- Piv txwv li, peb paub tias qhov ntev ntawm qhov hypotenuse ntawm daim duab peb sab yog 5 thiab qhov ntev ntawm ib sab yog 3, tab sis peb tsis paub meej txog qhov ntev ntawm peb sab. Hauv qhov no, peb paub tias peb tab tom nrhiav qhov ntev ntawm sab peb, thiab txij li peb paub qhov ntev ntawm lwm ob, peb tuaj yeem daws nws! Peb yuav ua haujlwm ntawm qhov teeb meem no nrog cov hauv qab no.
- Yog tias koj tsis paub qhov ntev ntawm ob sab, koj yuav tsum paub ib sab kom thiaj tuaj yeem siv Pythagorean Theorem. Kev ua haujlwm trigonometric yooj yim tuaj yeem pab koj yog tias koj paub ib sab ntawm daim duab peb sab uas tsis ntxeev.
Kauj Ruam 4. Plug qhov ob-tog qhov tseem ceeb uas koj twb paub rau hauv kab zauv
Txuas qhov ntev ntawm ob sab ntawm koj daim duab peb sab rau hauv kab zauv a2 + b2 = c2. Nco ntsoov tias a thiab b tsis yog sab nqes hav, thaum c yog hypotenuse.
Hauv peb qhov piv txwv, peb paub qhov ntev ntawm ib sab thiab qhov hypotenuse (3 & 5), yog li qhov sib npaug yuav dhau los 3² + b² = 5²
Kauj Ruam 5. Square
Txhawm rau daws koj qhov kev ua zauv, pib los ntawm kev txheeb xyuas ob tog uas paub. Xwb, yog tias koj pom qhov no yooj yim dua, koj tuaj yeem tso koj sab ntev sib npaug, thiab ua rau lawv tom qab.
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Hauv peb qhov piv txwv, peb yuav faib 3 thiab 5 kom peb tau txais
Kauj ruam 9. ua da
Kauj ruam 25.. Peb tuaj yeem sau qhov sib npaug li 9 + b² = 25.
Kauj Ruam 6. Tsiv qhov tsis paub sib txawv mus rau lwm sab ntawm qhov sib npaug
Yog tias xav tau, siv kev ua lej algebraic yooj yim los ua qhov tsis paub sib txawv txav mus rau lwm sab ntawm qhov kev ua zauv thiab cov xwm txheej ntawm ob qhov sib txawv mus rau lwm sab. Yog tias koj xav pom qhov ntev ntawm qhov hypotenuse, c twb tau nyob rau lwm sab ntawm qhov sib npaug, yog li koj tsis tas yuav ua dab tsi los txav nws.
Hauv peb qhov piv txwv, qhov sib npaug tam sim no yog 9 + b² = 25. Txhawm rau txav b², rho ob sab ntawm qhov kev ua zauv los ntawm 9, yog li qhov tshwm sim yog b² = 16
Kauj Ruam 7. Square cag ntawm ob sab ntawm qhov sib npaug
Tam sim no tsuas yog ib qho sib txawv sib luag ntawm ib sab thiab tus lej ntawm lwm qhov. Square cag ntawm ob sab kom pom qhov ntev ntawm sab tsis paub.
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Hauv peb qhov piv txwv, b² = 16, siv lub hauv paus square ntawm ob sab muab b = 4. Yog li, peb tuaj yeem hais tias qhov ntev ntawm qhov tsis paub sab ntawm daim duab peb sab yog
Kauj ruam 4..
Kauj Ruam 8. Siv Pythagorean Theorem los nrhiav ob sab ntawm daim duab peb sab txoj cai tseeb
Yog vim li cas Pythagorean Theorem tau siv dav rau niaj hnub no yog tias nws tuaj yeem siv tau rau ntau qhov xwm txheej uas siv tsis tau. Kawm kom paub txoj cai peb tog hauv lub neej tiag tiag - hauv txhua qhov xwm txheej uas ob yam khoom lossis kab ncaj ncaj ntsib lub kaum sab xis thiab cov khoom thib peb lossis kab koom nrog ob yam khoom lossis kab kab kab pheeb ces kaum, tom qab ntawd koj tuaj yeem siv Pythagorean Theorem los nrhiav qhov ntev ntawm sab lwm qhov, yog tias qhov ntev ntawm ob sab tau paub.
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Cia peb sim ua piv txwv tiag uas nyuaj me ntsis. Tus ntaiv leans tawm tsam ib lub tsev. Qhov kev ncua deb ntawm qab ntaiv mus rau phab ntsa yog 5 meters. Qhov siab ntawm tus ntaiv nce mus txog 20 meters. Tus ntaiv ntev npaum li cas?
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5 meters ntawm phab ntsa thiab siab 20 meters qhia peb qhov ntev ntawm ob sab ntawm daim duab peb sab. Txij li ntawm phab ntsa thiab hauv av (xav tias) tsim lub kaum sab xis thiab tus ntaiv tau teeb tsa kab pheeb ces kaum tiv thaiv phab ntsa, qhov kev npaj no tuaj yeem suav tias yog daim duab peb sab xis nrog sab ntev a = 5 thiab b = 20. Qhov ntev ntawm tus ntaiv yog qhov hypotenuse, yog li tus nqi c tsis paub. Cia peb siv Pythagorean Theorem:
- a² + b² = c²
- (5) ² + (20) ² = c²
- 25 + 400 = np
- 425 = np
- hauv paus (425) = c
- c = 20.6 ib. Qhov kwv yees ntev ntawm tus ntaiv yog 20.6m ib.
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Txoj Kev 2 ntawm 2: Xam Xaj Qhov Nruab Nrab Ntawm Ob Lub Ntsiab Lus hauv X-Y. Dav Hlau
Kauj Ruam 1. Nrhiav ob lub ntsiab lus hauv X-Y lub dav hlau
Pythagorean theorem tuaj yeem siv tau yooj yim los xam cov kab ncaj nraim nrug ntawm ob lub ntsiab lus hauv X-Y lub dav hlau. Txhua yam koj yuav tsum paub yog x thiab y tswj hwm ntawm ob lub ntsiab lus. Feem ntau, cov haujlwm no tau sau ua ke hauv daim ntawv (x, y).
Txhawm rau nrhiav qhov kev ncua deb ntawm ob lub ntsiab lus no, peb yuav txiav txim siab txhua lub ntsiab lus ua ib qho ntawm cov tsis yog txoj cai ntawm txoj cai peb tog. Ua li ntawd yuav ua kom yooj yim nrhiav qhov ntev ntawm ob sab a thiab b, thiab tom qab ntawd suav qhov hypotenuse c, uas yog qhov nrug nruab nrab ntawm ob lub ntsiab lus
Kauj Ruam 2. Kos koj ob lub ntsiab lus hauv daim duab
Hauv X-Y lub dav hlau tsis tu ncua, txhua lub ntsiab lus (x, y), x sawv cev rau kev sib koom ua ke kab rov tav thiab y sawv cev rau kev sib koom ua haujlwm ntsug. Koj tuaj yeem pom qhov kev ncua deb ntawm ob lub ntsiab lus yam tsis kos nws, tab sis ua li ntawd yuav muab cov duab pom uas koj tuaj yeem siv los saib yog tias koj cov lus teb raug.
Kauj Ruam 3. Nrhiav qhov ntev ntawm sab uas tsis nqes hav ntawm koj daim duab peb sab
Siv ob lub ntsiab lus raws li cov ces kaum ntawm daim duab peb sab uas nyob ib sab rau qhov hypotenuse, nrhiav qhov ntev ntawm ob sab a thiab b ntawm daim duab peb sab. Koj tuaj yeem ua qhov no siv cov duab lossis siv cov mis | x1 -x ib2| rau sab tav toj thiab | y1 -ib y2| rau sab ntsug, nrog (x1, y xub1) raws li thawj kis thiab (x2, y xub2) raws li lub ntsiab lus thib ob.
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Cia peb ob lub ntsiab lus yog (6, 1) thiab (3, 5). Qhov ntev ntawm kab rov tav ntawm peb daim duab peb sab yog:
- | x1 -x ib2|
- |3 - 6|
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| -3 | =
Kauj ruam 3.
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Qhov ntev ntawm sab ntsug yog:
- | y1 -ib y2|
- |1 - 5|
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| -4 | =
Kauj ruam 4.
- Yog li, hauv peb daim duab peb sab xis, sab a = 3 thiab sab b = 4.
Kauj Ruam 4. Siv Pythagorean Theorem los nrhiav qhov ntev ntawm cov hypotenuse
Qhov kev ncua deb ntawm ob lub ntsiab lus yog qhov ntev ntawm hypotenuse ntawm daim duab peb sab uas nws ob sab koj nyuam qhuav pom. Siv Pythagorean Theorem los nrhiav tus hypotenuse, qhov twg a yog qhov ntev ntawm thawj sab thiab b yog qhov ntev ntawm ob sab.
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Hauv peb qhov piv txwv, peb tab tom siv cov ntsiab lus (3, 5) thiab (6, 1) uas nws sab ntev yog 3 thiab 4, yog li peb tuaj yeem pom qhov hypotenuse raws li hauv qab no:
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- (3) ²+(4) ² = c²
- c = hauv paus (9+16)
- c = hauv paus (25)
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c = 5. Qhov nrug nruab nrab ntawm (3, 5) thiab (6, 1) yog
Kauj ruam 5..
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Lub tswv yim
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Qhov hypotenuse yog ib txwm:
- rov qab lub kaum sab xis (tsis kov lub kaum sab xis)
- sab ntev tshaj plaws hauv daim duab peb sab
- hu ua c hauv Pythagorean theorem
- hauv paus (x) txhais tau tias yog lub hauv paus cag ntawm x.
- Nco ntsoov xyuas koj cov lus teb tas li. Yog tias koj cov lus teb zoo li tsis raug, sim dua thiab sim dua.
- Yog tias daim duab peb sab tsis yog daim duab peb sab xis, koj xav tau cov ntaub ntawv ntxiv, tsis yog qhov ntev ntawm ob sab nkaus xwb.
- Lwm txoj hauv kev ntawm kev tshuaj xyuas - sab ntev tshaj plaws yog qhov txawv ntawm lub kaum sab xis loj tshaj plaws thiab sab luv tshaj yog lub kaum sab xis me tshaj plaws.
- Cov lej yog tus yuam sij rau sau qhov raug qhov raug rau a, b, thiab c. Yog tias koj tab tom ua haujlwm ntawm cov teeb meem dab neeg, nco ntsoov sau qhov teeb meem hauv daim duab ua ntej.
- Yog tias koj tsuas paub qhov ntev ntawm ib sab, Pythagorean Theorem tsis ua haujlwm. Sim siv trigonometry (sin, cos, tan) lossis 30-60-90 / 45-45-90 piv.
