Dudzirazivo raPythagoras

Nharaunda yeskweya rine divi c, muzendami , yakaenzana nekubatanidzwa kwenharaunda dzemaskweya maviri (a, b) ane mativi ari pamakumbo maviri egonyonhatu yakarurama

Muchidzidzo pimanyika, Dudzirazivo raPythagoras, itsazaniso yakanyorwa naPythagoras inotaura hukama huripo pakati pemativi matatu egonyonhatu yakarurama. Zvichinyorwa sehukuru hwenharaunda, dudzirazivo iri rinoyorwa sezvizvi:

Nharaunda yeskweya rine divi muzendami yakaenzana nekubatanidzwa kwenharaunda dzemaskweya maviri ane mativi ari pamakumbo maviri egonyonhatu yakarurama - (apa kuchirehwa makumbo anosangana pagonyo rakarurama).

Dudzirazivo iri rinonyorwa setsazaniso inotaura hukama pakati pehurefu hwemativi a, b and c, inonzi Tsazaniso yaPythagoras:

$a^{2}+b^{2}=c^{2}\!\,$

Apa c anomirira hurefu hwemuzendami, a na b vachimirira hurefu hwemakumbo maviri anosangana pagonyo rakarurama.

Dudzirazivo iri rakapiwa zita remurume wekuGreece anonzi Pythagoras anova ndiye anonzi akatanga kuwona hukama pakati pemativi egonyonhatu yakarurama. Asiwo kunonzi magodobori emasvomhu ekuBabylon akange achiziva zvohukama uhwu Pythagoras asati azviziva. Pari zvino hakuchina humbowo hunotsigira kuti vaBabylon vakanga vachiziva dudzirazivo raPythagoras kana kuti vakanga vamboshandisa ruzivo rwunotaurwa naPythagoras.

Mamwe manyorerwo

Sekutaurwa kwamboitwa kana c achimirira hurefu hwemuzendami, a na b vachimirira hurefu hwemamwe mativi, dudzirazivo raPythagoras rinonyorwa setsazaniso yaPythagoras sezvizvi:

$a^{2}+b^{2}=c^{2}\,$

Kana hurefu hwemativi a na b huchizivikanwa, hurefu hwedivi c hunowanikwa sezvizvi:

$c={\sqrt {a^{2}+b^{2}}}.\,$

Kana hurefu hwemuzendami c nerimwe divi (a kana b) huchizivikanwa, tinenge toti hurefu hwerimwe divi hunowanikwa netsazaniso idzi:

$a={\sqrt {c^{2}-b^{2}}}.\,$

kana

$b={\sqrt {c^{2}-a^{2}}}.\,$

Dudzirazivo raPythagoras rinopa hukama hwemativi egonyonhatu yakarurama nenzira yakareruka, zvokuti kana hurefu hwemativi maviri huchizivikanwa, hurefu hwedivi rechitatu hauchanetse kuwana. Nekumwe kutaura kunonongedza zviripo padudzirazivo, muzendami wakareba kudarika divi rimwe nerimwe asi muzendami mupfupi pane kubatanidzwa kwehurefu hwemativi maviri.

Humbowo hunotsigira

Humbowo hwegonyonhatu dzakafanana

Humbowo hunoshanda negonyonhatu dzakafanana

Humbowo huri pano hunoshandisa muzasedimbu uri pakati pemativi egonyonhatu dzakafanana, izvi zvichireva kuenzana kwerheshiyo dzemativi egonyonhatu dzakafanana, zvisinei kuti gonyonhatu idzi hadzina kuenzana.

Ngatiti ABC rive gonyonhatu yakarurama ine gonyo rakarurama pana C, sezviri kutaridzwa pamufananidzo. Ngatitarei mutsetse kubva pana C, uchinorurama pahwaro AB papoyindi H. Poyindi H inokamura muzendami c kuita zvikamu d na e. Gonyonhatu idzva ACH rakafanana negonyonhatu ABC nokuti dzose dzine gonyo re90^o, uye dzine gonyo rakaenzana pana A. Izvi zvinoreva kuti gonyo rechitatu rakaenzana mugonyonhatu dziri mbiri, iri gonyo riri kudanwa kunzi θ pamufananidzo. Nenzira imwechete yemafungiro tinoona kuti gonyonhatu CBH rakafanana negonyonhatu ABC. Kufanana kwegonyonhatu idzi kunoreva kuti panokwanisa kunyorwa tsazaniso dzinotevera:

${\frac {BC}{AB}}={\frac {BH}{BC}}{\text{ and }}{\frac {AC}{AB}}={\frac {AH}{AC}}.\,$

Tsazaniso rekutanga rinoenzanisa cosine dzemakonyo ane zita θ, tsazaniso rechipiri rinoenzanisa sine dzemakonyo.

Rheshiyo idzi dzinonyorwa sezvizvi:

${BC}^{2}={AB}\times {BH}{\text{ and }}{AC}^{2}={AB}\times {AH}.\,$

Tichiwedzanisa tsazaniso mbiri idzi tinowana,

${BC}^{2}+{AC}^{2}={AB}\times {BH}+{AB}\times {AH}={AB}\times ({AH}+{BH})={AB}^{2},\,\!$

zvakanyorwa pamusoro kana zvarerutswa zvinopa dudzirazivo raPythagoras:

${BC}^{2}+{AC}^{2}={AB}^{2}\ .\,\!$