Distanțe în cub

Fie cubul ABCDA’B’C’D’ de latură a. Aflați: a) d(A’; D); b)

d(A;BB’);d(A’;CC’);d(C;AC’);d(A’;BD); c) d(D’; (ABC)); d(A; (A’BD)); d) d(A’D’;AD); d(A’D’;BC); d(A’D;

B’C); e) d(A’B’; (DCC’)); f) d((A’AD); (B’C’C)).

1.a)

d(A’; D) = ?

1.a)

d(A’; D) = A’D =

2aa2aa 222

1.b1)

d(A; BB’) = ?

1.b1)

AB BB’ => d(A; BB’) = AB = a

1.b2)

d(A’; CC’) = ?

1.b2)

d(A’; CC’) = A’C’

CC’ (A’B’C’), A’C’ (A’B’C’) => CC’ A’C’

1.b2)

d(A’; CC’) = A’C’ = a

A’B’C’, (m<(B’) = 90o) => A’C’ = a

2

2

1.b3)

d(C; AC’) = ?

1.b3)

CE AC’ => d(C; AC’) = CE

1.b3)

CE = hACC’; ACC’ = dreptunghic

d(C; AC’) = CE = (AC . CC’)/AC’ =

36a

1.b4)

d(A’; BD) = ?

1.b4)

A’O DB => d(A’; DB) = A’O

1.b4)

DB=A’D=A’B= d(A’;BD) = A’O = hA’BD

= 26a

2a => A’BD = echilateral

1.c1)

d(D’; (ABC)) = ?

1.c1)

D’D (ABC) => d(D’; (ABC)) = D’D = a

1.c2)

d(A; (A’BD)) = ?

1.c2)

AH (A’BD) => d(A; (A’BD)) = AH

1.c2)

AH = hA’AO; A’AO = dreptunghic

d(A; (A’BD)) = AH = (AO . AA’)/OA’ =

33a

1.d1)

d(A’D’; AD) = ?

1.d1)

A’D’ AD; AA’ AD; AA’ A’D’

d(A’D’; AD) = AA’ = a

1.d2)

d(A’D’; BC) = ?

1.d2)

A’D’ BC; A’B BC; A’B A’D’

d(A’D’; BC) = A’B =

2a

d(A’D; B’C) = ?

1.d3)

1.d3)

A’D B’C; A’B’ A’D; A’B’ B’C

d(A’D; B’C) = A’B’ = a

d(A’B’; (DCC’)) = ?

1.e)

1.e)

A’B’ (DCC’); B’C’ A’B’; B’C’ (DCC’)

d(A’B’; (DCC’) = B’C’ = a

d((A’AD); (B’C’C)) = ?

1.f)

1.f)

(A’AD) (B’C’C); AB (A’AD); AB (B’C’C)

d(A’AD); (B’C’C)) = AB = a