Twierdzenie 1
Stosunek pól figur podobnych równa się kwadratowi
skali podobieństwa.
Przykład 1
Na rysunku dane są dwa trapezy podobne, których pola
pozostają w stosunku
. Korzystając z danych z rysunku, oblicz ich
pola.
![](data:image/png;base64,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Wyliczamy skalę podobieństwa tych trapezów wiedząc, że
stosunek pól jest równy skali podniesionej do kwadratu.
![](data:image/png;base64,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)
, bo k >0.
Następnie możemy policzyć długość wysokości większego
trapezu.
![](data:image/png;base64,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)
Teraz już liczymy pola trapezów
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Odp. Pola są równe odpowiednio 8 cm i 32cm.
Znane jest nam twierdzenie Pitagorasa o treści:
„Jeżeli trójkąt jest prostokątny, to kwadrat długości
przeciwprostokątnej jest równy sumie kwadratów długości przyprostokątnych.”
Możemy je przeredagować i sformułować też w ten sposób:
„Jeżeli trójkąt jest prostokątny i na bokach tego trójkąta
zbudujemy figury podobne, to pole figury zbudowanej na przeciwprostokątnej jest
równe sumie pól figur zbudowanych na przyprostokątnych.”
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Dowód tego twierdzenia możemy znaleźć w podrozdziale
„Zastosowanie pojęcia pola w dowodzeniu twierdzeń”.