数学试题讲解

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如图, 在正四棱柱

A B C D - A_{1} B_{1} C_{1} D_{1}

中,

A B = 2, A A_{1} = 4

. 点

A_{2}, B_{2}, C_{2}, D_{2}

分别在棱

A A_{1}, B B_{1}, C C_{1}, D D_{1}

上,

A A_{2} = 1, B B_{2} = D D_{2} = 2, C C_{2} = 3

.

(1)证明:

B_{2} C_{2} / / A_{2} D_{2}

;
(2) 点

P

在棱

B B_{1}

上, 当二面角

P - A_{2} C_{2} - D_{2}

为

150^{\circ}

时, 求

B_{2} P

.