Twierdzenia przedstawione w tym temacie mogą w pewnych
przypadkach ocenić, czy kąt między przecinającymi się w przestrzeni prostymi
jest prosty.
Twierdzenie 1: Jeśli prosta l przebija płaszczyznę π i nie jest do niej prostopadła oraz prosta m
leżąca na płaszczyźnie π jest prostopadła do rzutu prostokątnego l1 prostej I na płaszczyznę π i przecina prostą l, to prosta m jest
prostopadła do prostej l.
Twierdzenie 2: Jeśli prosta I przebija płaszczyznę π i nie jest do niej prostopadła oraz prosta m
leżąca na płaszczyźnie π jest prostopadła do prostej I, to jest prostopadła do
rzutu prostokątnego l1 prostej I na płaszczyznę π.
Powyższe dwa twierdzenia zapisując jednocześnie otrzymujemy twierdzenie
o trzech prostych prostopadłych
Twierdzenie o trzech prostych prostopadłych (Tw.3): Jeśli
prosta l przebija płaszczyznę π i nie jest do niej prostopadła, prosta l1 jest rzutem prostokątnym prostej I na
płaszczyznę π, prosta m leży na
płaszczyźnie π i przecina prostą l, to prosta m jest
prostopadła do prostej I wtedy i tylko wtedy, gdy jest prostopadła do prostej l1.
Przykład
W trójkącie równoramiennym ABC leżącym na
płaszczyźnie, gdzie ǀABǀ=ǀACǀ=25 i ǀBCǀ=20. W punkcie D będącym środkiem boku BC
poprowadzono prostą prostopadłą do płaszczyzny. Punkt E jest punktem na tej
prostej takim, że ǀEBǀ= 10. Oblicz pole trójkąta AEB.
Rozwiązanie:Rysunek:
![](data:image/png;base64,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)
Zauważamy, że AD jest wysokością trójkąta ADC, z twierdzenia
Pitagorasa dla trójkąta ADC otrzymujemy:
→ ǀADǀ = ![](data:image/png;base64,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)
Pole ADC
=![](data:image/png;base64,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)
Zadania do zrobienia
1. W trójkącie różnobocznym ABC punkt E jest środkiem
odcinka BC. Odcinek AD jest prostopadły do płaszczyzny (ABC). Czy prosta DE
jest prostopadła do prostej BC? Odpowiedź uzasadnij.
Odp. Nie.
2. W trójkącie prostokątnym ABC przyprostokątne mają
długość: |AC| = 15 cm, |BC| = 20 cm. Odcinek DC jest prostopadły do płaszczyzny
(ABC) i ma długość 22,5 cm. Oblicz
wysokość trójkąta ABD poprowadzoną na bok AB.
Odp. 25,5 cm.
3. W trójkącie ABC boki mają długość: |AB| = 60 cm, |AC| =
|BC| = 50 cm. Odcinek AD jest prostopadły do płaszczyzny (ABC). Odległość punktu D od
prostej BC jest równa 52 cm. Oblicz odległość punktu D od płaszczyzny (ABC).
Odp. 20 cm.