In the cube ABCD -- a'b'c'd ', O is the center of the square ABCD on the bottom, and M is the midpoint of the line a'b Verification: plane a'bd ⊥ plane a'acc ' emergency

In the cube ABCD -- a'b'c'd ', O is the center of the square ABCD on the bottom, and M is the midpoint of the line a'b Verification: plane a'bd ⊥ plane a'acc ' emergency

It is proved that because the bottom of a'a ⊥ is ABCD, and the line BD belongs to the plane ABCD,
So line segment a'a ⊥ line segment BD
Because AC and BD are diagonals of the bottom of the cube
So AC ⊥ BD
Because AC ⊥ BD, a'a ⊥ BD, and a'a ∩ AC at point a,
So BD is perpendicular to plane aa'cc '
Because BD ∈ plane a'bd

The length of each side edge and bottom side of a straight triangular prism is a, and point D is any point on CC '. Connect a'B, BD, a'd, ad, and find the volume of a-a'bd?

∵ abc-a 'B' C 'is a straight triangular prism, ∵ AC ⊥ AA', AA ∥ CD,
The area of Δ AA ′ d = the area of Δ AA ′ C = AC × AA ′ / 2 = a ^ 2 / 2
∵ abc-a ′ B ′ C ′ is a straight triangular prism, the distance from B to plane AA ′ d = the distance from B to AC = (√ 3 / 2) AB = √ 3A / 2
The volume of triangular pyramid a-a'bd = (1 / 3) (a ^ 2 / 2) (√ 3A / 2) = √ 3A ^ 3 / 12