题干

如图,在四棱锥P﹣ABCD中,PD⊥平面ABCD,底面ABCD是菱形,∠BAD=60°,AB=2,PD= 6 ,O为AC与BD的交点,E为棱PB上一点.

(Ⅰ)证明:平面EAC⊥平面PBD;

(Ⅱ)若PD∥平面EAC,求三棱锥P﹣EAD的体积.

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答案(点此获取答案解析)

(Ⅰ)证明:∵PD⊥平面ABCD,AC⊂平面ABCD,

∴AC⊥PD.∵四边形ABCD是菱形,∴AC⊥BD,

又∵PD∩BD=D,AC⊥平面PBD.

而AC⊂平面EAC,∴平面EAC⊥平面PBD.

(Ⅱ)解:∵PD∥平面EAC,平面EAC∩平面PBD=OE,

∴PD∥OE,

∵O是BD中点,∴E是PB中点.

取AD中点H,连结BH,∵四边形ABCD是菱形,∠BAD=60°,

∴BH⊥AD,又BH⊥PD,AD∩PD=

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    Chinese taikonauts, Jing Haipeng, 50, and Chen Dong, 37, were blasted off into space onboard Shenzhou-11 at 7:30 am Monday and will spend 30 days in the Chinese space laboratory Tiangong-2.The launch marks a key step toward China's plan to eventually operate a permanent space station. The successful launch of the Shenzhou-11 spacecraft is another step forward to put China among leading players in space technology, said Alexander Zheleznyakov, a Russian expert on history of space flights. China's experimental space lab will help provide solutions for spacecraft of different functions to approach and anchor, and for a long-term operation of life support system, said Zheleznyakov. Shenzhou-11 is scheduled to anchor on Wednesday with Tiangong-2, which is part of China's plan to build a permanent space station by 2022.

    China can now test technologies for cargo spacecraft anchoring, life support system operation and water recycling to ensure a long-term continuous operation of its space station in the future with less dependence on renewal from the Earth, he said. If all goes well, China will launch the unpiloted Tianzhou-1 cargo ship next spring to autonomously tie up with Tiangong-2. Tianzhou-1 will be capable of automatically transferring rocket fuels, a vital requirement for space station grouping and maintenance, according to a report by Columbia Broadcasting System (CBS).

    "That will further their anchoring abilities needed for the larger space station," Johnson Freese was quoted." Tiangong-2 is supposed to be able to stay in orbit for two years or longer, so that's taking them (Chinese) really close to 2019 or so. I think this will be their last big technology test phase before going to their large space station," said Freese.

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