问题 解答题

文(12分)已知四棱锥P-ABCD,PB⊥AD,侧面PAD为边长等于2的正三角形,底面ABCD为菱形,侧面PAD与底面ABCD所成的二面角为120°.(1)求点P到平面ABCD的距离;(2)求PD与AB所成角的大小;(3)求二面角A—PB—C的大小.

答案

(1)(2)(3)

(1)作PO⊥平面ABCD于O,则PO⊥AD,又∵PB⊥AD,

∴AD⊥平面POB,连OB交AD于E,则PE⊥AD,BE⊥AD,

得∠PEB为二面角P-AD-B的平面角.∴∠PEB=120°,

在边长为2正△PAD中,易得AE=,∴为所求;

(2)易证Rt△PAE≌Rt△BAE(直角边、斜边).∴BE=PE=,∴PB=3.又在Rt△PBC中.∵AB∥DC,∴PD与AB所成角即为PD与DC所成角.在△PDC中,由余弦定理得.∴PD与AB所成角大小为.

(3)取PB中点G及PC中点F,则GF∥BC,而BC⊥PB,∴GF⊥PB;又∵AP=AB,∴AG⊥PB,于是∠AGF为所求平面角.由(2)所证知PE=BE,∴∠PEG=60°,,∴Rt△GAE中, ,∴.

解法2:建立如图坐标系,则,先证明,从而知B,

G,A,C.然后由,如所成的角即为所求平面角.∵,∴平面角.

注:(2)题中可由.

阅读理解

Have you ever lost your temper? Did you yell or scream at those around you, even people you like or love? Did you want to break something or hit someone?

Everyone gets angry. We have lots of emotions. At different times, we may be happy, sad or jealous.    小题1:  .

It’s perfectly okay to be angry at time(偶尔)--- in fact, it’s important to get angry sometimes. Anger can even be a good thing. When we are treated unfairly, anger can help us stand up for ourselves.

__小题2:___ some people keep their anger buried deep inside. If you do this, you might get a headache or your stomach might start to hurt. You may just feel crummy (糟糕的) about yourself or start to cry. It’s not good to hide your anger, so you should find a way to let it out without hurting yourself or others.

When you start to feel angry, you can count to 10, draw a picture of anger, play a video game, run as fast as you can or do something active. An effective way is to talk to a friend you can trust.   小题3:  .

小题4:  Instead, admit to yourself that you are angry and try to figure out why you are angry and what you can do to keep the situation from happening again.

Never getting angry is impossible. But, you should always remember that how you act when you’re angry can make the situation better or worse.  

小题5: Take charge of it!

A. Once you talk about anger, those bad feelings usually start to go away.

B. Taking it out on others never solves anything.

C. Anger is just another way we feel.

D. Here is an important way to deal with anger.E.  But anger must be released in the right way.

F.  Every coin has two sides.

G.  Don’t let anger be the boss of you.

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