Ringa mabviro eDudzirazivo raPythagoras

[[File:Pythagorean.svg|thumb|Nharaunda ye[[skweya]] rine divi '''c''', [[muzendami]] , yakaenzana nekubatanidzwa kwe[[nharaunda]] dzema[[skweya]] maviri ('''a, b''') ane mativi ari pamakumbo maviri egonyonhatu yakarurama]]

Muchidzidzo [[pimanyika]], [[Dudzirazivo raPythagoras]], i[[tsazaniso]] yakanyorwa na[[Pythagoras]] inotaura hukama huripo pakati pemativi matatu e[[gonyonhatu yakarurama]]. Zvichinyorwa sehukuru hwe[[nharaunda]], dudzirazivo iri rinoyorwa sezvizvi:


''Nharaunda ye[[skweya]] rine divi [[muzendami]] yakaenzana nekubatanidzwa kwe[[nharaunda]] dzema[[skweya]] maviri ane mativi ari pamakumbo maviri egonyonhatu yakarurama - (apa kuchirehwa makumbo anosangana pa[[gonyo]] raka[[rurama]]).''


Dudzirazivo iri rinonyorwa setsazaniso inotaura hukama pakati pe[[hurefu]] hwemativi ''a'', ''b'' and ''c'', inonzi ''Tsazaniso yaPythagoras'':


:<math>a^2 + b^2 = c^2\!\,</math>

Apa '''c''' anomirira hurefu hwe[[muzendami]], '''a''' na '''b''' vachimirira hurefu hwemakumbo maviri anosangana pagonyo rakarurama.

Dudzirazivo iri rakapiwa zita remurume weku[[Greece]] anonzi Pythagoras anova ndiye anonzi akatanga kuwona hukama pakati pemativi egonyonhatu yakarurama. Asiwo kunonzi magodobori emasvomhu eku[[Babylon]] akange achiziva zvohukama uhwu Pythagoras asati azviziva. Pari zvino hakuchina humbowo hunotsigira kuti va[[Babylon]] 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:

: <math>a^2 + b^2 = c^2\, </math>

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

: <math> c = \sqrt{a^2 + b^2}. \,</math>

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

: <math>a = \sqrt{c^2 - b^2}. \,</math>

kana 

: <math>b = \sqrt{c^2 - a^2}. \,</math>

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

==Humbowo hunotsigira==


'''Humbowo hwe[[gonyonhatu]] dzakafanana'''

[[File:Pythagoras similar triangles simplified.svg|thumb|Humbowo hunoshanda negonyonhatu dzakafanana]]
Humbowo huri pano hunoshandisa [[muzasedimbu]] uri pakati pemativi egonyonhatu dzakafanana, izvi zvichireva kuenzana kwe[[rheshiyo]] dzemativi egonyonhatu dzakafanana, zvisinei kuti gonyonhatu idzi hadzina kuenzana.

Ngatiti ''ABC'' rive [[gonyonhatu yakarurama]] ine gonyo raka[[rurama]] pana '''C''', sezviri kutaridzwa pamufananidzo. Ngatitarei mutsetse kubva pana ''C'', uchino[[rurama]] pa[[hwaro]] '''AB''' pa[[poyindi]] '''H'''. Poyindi ''H'' inokamura [[muzendami]] '''c'''  kuita zvikamu  '''d'''  na '''e'''. Gonyonhatu idzva ''ACH'' [[Kufanana_(pimanyika)|rakafanana]] negonyonhatu ''ABC'' nokuti dzose dzine gonyo re90<sup>o</sup>, 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:

:<math> \frac{BC}{AB}=\frac{BH}{BC} \text{ and } \frac{AC}{AB}=\frac{AH}{AC}.\,</math>

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

Rheshiyo idzi dzinonyorwa sezvizvi:

:<math>{BC}^{2}={AB}\times {BH} \text{ and }{AC}^{2}={AB}\times {AH}. \,</math>

Tichiwedzanisa tsazaniso mbiri idzi tinowana,

:<math>{BC}^{2}+{AC}^{2}={AB}\times {BH}+{AB}\times {AH}={AB}\times({AH}+{BH})={AB}^{2} ,\,\!</math>

zvakanyorwa pamusoro kana zvarerutswa zvinopa [[dudzirazivo raPythagoras]]:

:<math>{BC}^{2}+{AC}^{2}={AB}^{2} \ .\,\!</math>

[[category:Pimanyika]]