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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 = a 2 + a 2 = 2a 2 = a 2

1.b1)

d(A; BB’) = ?

1.b1)

AB ⊥ BB’ => d(A; BB’) = AB = a

1.b2)

d(A’; CC’) = ?

1.b2)

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

d(A’; CC’) = A’C’

1.b2)

∆A’B’C’, (m<(B’) = 90o) => A’C’ = a 2

d(A’; CC’) = A’C’ = a 2

1.b3)

d(C; AC’) = ?

1.b3)

CE ⊥ AC’ => d(C; AC’) = CE

1.b3)

CE = h∆ACC’; ∆ACC’ = dreptunghic

a 6
d(C; AC’) = CE = (AC . CC’)/AC’ =
3

1.b4)

d(A’; BD) = ?

1.b4)

A’O ⊥ DB => d(A’; DB) = A’O

1.b4)

DB=A’D=A’B= a 2 => ∆A’BD = echilateral

a 6
d(A’;BD) = A’O = h∆A’BD =
2

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 = h∆A’AO; ∆A’AO = ∆ dreptunghic

a 3
d(A; (A’BD)) = AH = (AO . AA’)/OA’ =
3

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 = a 2

1.d3)

d(A’D; B’C) = ?

1.d3)

A’D | | B’C; A’B’ ⊥ A’D; A’B’ ⊥ B’C

d(A’D; B’C) = A’B’ = a

1.e)

d(A’B’; (DCC’)) = ?

1.e)

A’B’ | | (DCC’); B’C’ ⊥ A’B’; B’C’ ⊥ (DCC’)

d(A’B’; (DCC’) = B’C’ = a

1.f)

d((A’AD); (B’C’C)) = ?

1.f)

(A’AD) | | (B’C’C); AB ⊥ (A’AD); AB ⊥ (B’C’C)

d(A’AD); (B’C’C)) = AB = a

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Distante in cub clasa 8

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