In the cube ABCD-A ′ B ′ C ′ D ′ with edge length 1, AC ′ is diagonal, M. N is the midpoint of BB ′, B ′, C ′, P is the midpoint of Mn (1) Finding the tangent of the angle between DP and ABCD (2) Find the tangent of the angle between DP and AC '

In the cube ABCD-A ′ B ′ C ′ D ′ with edge length 1, AC ′ is diagonal, M. N is the midpoint of BB ′, B ′, C ′, P is the midpoint of Mn (1) Finding the tangent of the angle between DP and ABCD (2) Find the tangent of the angle between DP and AC '

Taking the surface cc1d1d as the contact surface, a cube cdefc1d1e1f1 with the same size is superimposed
Make PQ ⊥ BC to Q through P and connect DQ
It is easy to get the ABCD of PQ ⊥ surface, so PQ ⊥ DQ
PQ = 3 / 4, DQ = root 5 / 2, then the tangent of the angle = PQ / DQ = 3 root 5 / 10
DP = root 29 / 4
DF1 / / AC1 f1p = 5 radical 2 / 4 DF1 = radical 3
Then the cosine of the angle formed by DP AC1 can be obtained by the cosine theorem, and the root sign of the tangent value is 4839 / 27

In ABCD -- a'b'c'd ', where p is on AC, q is on BC', and AP = BQ = a
(1) (2) verification: PQ ⊥ ad

Make QE ⊥ BC, connect PE. 1, AC = √ 2a, & nbsp; CP = √ 2a-a & nbsp; & nbsp; QE = be = √ 2A / 2 & nbsp; CE = a - √ 2A / 2cp: CA = 1 - √ 2 / 2ce: CB = 1 - √ 2 / 2cp: CA = Ce: cbpe / / AB & nbsp; PE ⊥ BCPE = CE = a - √ 2A / 2tan (QPE) = QE / PE = √ 2 + 1 ∠ QPE = arctan (√ 2 + 1) line PQ and

In the cube abcd-a'b'c'd ', find the angle between Da' and AC
Figure matching

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The cubes abcd-a1b1c1d1, e and F are the midpoint of Aa1 and CC1, respectively. P is the moving point (including the endpoint) on CC1. Passing through the points e, D and P are the cross sections of the cube. If the cross section is a quadrilateral, then the trajectory of P is ()
A. Segment c1fb. Segment CFC. Segment CF and point C1d. Segment C1F and point C

As shown in the figure, the intersection of de ‖ plane bb1c1c, plane dep and plane bb1c1c1c, PM ‖ ed, connects em. it is easy to prove that MP = ed, ‖ MP ‖ ed, then M can still form a quadrilateral when it reaches B1, that is, P to F. P is between C1F, which does not meet the requirements. P to point C1 can still form a quadrilateral