Definicja 1
Podobieństwem nazywamy takie przekształcenie płaszczyzny, które dowolnym
dwóm różnym punktom A, B płaszczyzny przyporządkowuje takie punkty
,
dla których
, gdzie k jest ustaloną (dla danego podobieństwa)
liczbą dodatnią. Liczbę k nazywamy skalą
podobieństwa.
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Definicja 2
Figurami
podobnymi nazywamy takie dwie figury geometryczne F i
, dla których istnieje
podobieństwo przekształcające figurę F na figurę
. Podobieństwo figur F i
oznaczamy symbolicznie
.
Przekształcenie geometryczne, które nie zmienia odległości
między punktami, nazywamy izometrią.
Możemy o niej powiedzieć, że zachowuje wielkość i kształt figur. Przykładami
izometrii są przesunięcie równoległe, symetria osiowa czy symetria środkowa.
Zatem izometria jest podobieństwem o
skali 1.
Przykład 1
Wyznaczmy skalę podobieństwa, w której obrazem odcinka
|AB|=25cm jest odcinek ![](data:image/png;base64,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)
Skala podobieństwa, jest równa stosunkowi długości odcinka
do długości odcinka |AB|. Zatem:
![](data:image/png;base64,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)
Odp. Skala podobieństwa jest równa
.
Zadania do zrobienia
1. Obrazem półkola
w podobieństwie o skali
jest półkole
, którego obwód wynosi
. Oblicz długość
promienia półkola F.
Odp.
cm
2. Długości boków jednego pięciokąta mają
się do siebie jak
, a jego obwód jest równy
. Oblicz długość boków
drugiego pięciokąta, będącego obrazem pierwszego w podobieństwie o skali
.
Odp. ![](data:image/png;base64,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)