The document discusses distances between points and planes in a cube with side length a. It asks to find: 1) The distance between points A' and D is √2a^2 = a√2 2) Various distances between points and planes in the cube, which are all some form of a, a^2, or a√3 3) Whether lines are parallel or perpendicular to determine distances between lines and planes. All distances are expressed in terms of the side length a.

1. 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)).

2. 1.a)
d(A’; D) = ?

3. 1.a)
d(A’; D) = A’D = a 2 + a 2 = 2a 2 = a 2

4. 1.b1)
d(A; BB’) = ?

5. 1.b1)
AB ⊥ BB’ => d(A; BB’) = AB = a

6. 1.b2)
d(A’; CC’) = ?

7. 1.b2)
CC’ ⊥ (A’B’C’), A’C’ ⊂ (A’B’C’) => CC’ ⊥ A’C’
d(A’; CC’) = A’C’

8. 1.b2)
∆A’B’C’, (m<(B’) = 90o) => A’C’ = a 2
d(A’; CC’) = A’C’ = a 2

9. 1.b3)
d(C; AC’) = ?

10. 1.b3)
CE ⊥ AC’ => d(C; AC’) = CE

11. 1.b3)
CE = h∆ACC’; ∆ACC’ = dreptunghic
a 6
d(C; AC’) = CE = (AC . CC’)/AC’ =
3

12. 1.b4)
d(A’; BD) = ?

13. 1.b4)
A’O ⊥ DB => d(A’; DB) = A’O

14. 1.b4)
DB=A’D=A’B= a 2 => ∆A’BD = echilateral
a 6
d(A’;BD) = A’O = h∆A’BD =
2

15. 1.c1)
d(D’; (ABC)) = ?

16. 1.c1)
D’D ⊥ (ABC) => d(D’; (ABC)) = D’D = a

17. 1.c2)
d(A; (A’BD)) = ?

18. 1.c2)
AH ⊥ (A’BD) => d(A; (A’BD)) = AH

19. 1.c2)
AH = h∆A’AO; ∆A’AO = ∆ dreptunghic
a 3
d(A; (A’BD)) = AH = (AO . AA’)/OA’ =
3

20. 1.d1)
d(A’D’; AD) = ?

21. 1.d1)
A’D’ | | AD; AA’ ⊥ AD; AA’ ⊥ A’D’
d(A’D’; AD) = AA’ = a

22. 1.d2)
d(A’D’; BC) = ?

23. 1.d2)
A’D’ | | BC; A’B ⊥ BC; A’B ⊥ A’D’
d(A’D’; BC) = A’B = a 2

24. 1.d3)
d(A’D; B’C) = ?

25. 1.d3)
A’D | | B’C; A’B’ ⊥ A’D; A’B’ ⊥ B’C
d(A’D; B’C) = A’B’ = a

26. 1.e)
d(A’B’; (DCC’)) = ?

27. 1.e)
A’B’ | | (DCC’); B’C’ ⊥ A’B’; B’C’ ⊥ (DCC’)
d(A’B’; (DCC’) = B’C’ = a

28. 1.f)
d((A’AD); (B’C’C)) = ?

29. 1.f)
(A’AD) | | (B’C’C); AB ⊥ (A’AD); AB ⊥ (B’C’C)
d(A’AD); (B’C’C)) = AB = a