这是二零二三年普通高等学校招生全国统一考试英语科听力部分。 could you please tell me how i can get to the student dorm building sure turn left when you walk out of this register office pass the bookstore and you'll see it we couldn't ask for a better day could we barbara, i know steve there isn't a cloud in the sky i love this time of year me too the flowers are beautiful but calling for rain though we got any plans this weekend we might start with our vegetable garden then the barbecue sounds fun but i still want to go fishing with george lisa i saw jack this morning you mean the former designer of our company what's he doing these days he said he's got a new job as an art reporter do you often go to the student union andrew pretty much once in a while i meet friends there and i spend a lot of time organizing activities in different clubs or associations what do you think terry well, it has one more bedroom than the last flat and the sitting room is big but there is not enough cupboard space in the kitchen it's cheaper than the last one we saw and it is in a good neighborhood well, maybe we'll have to see if the agent has anything else to show us what do you think terry well, it has one more bedroom than the last flat and the sitting room is big but there is not enough cupboard space in the kitchen it's cheaper than the last one we saw and it is in a good neighborhood well, maybe we'll have to see if the agent has anything else to show us thank you mrs johnson for what you've shared with our listeners now please tell us do you sometimes feel worried about your daughter now that she's grown up oh, i worry about her like all parents do after she graduated from medical school, she went to volunteer medical service in africa wow, that's okay, if communication was fine, but if there was no internet or telephone connection we could sometimes go two weeks without getting in touch with each other that often worries me thank you missus johnson for what you've shared with our listeners now, please tell us do you sometimes feel worried about your daughter now that she's grown up, oh, i worry about her like all parents do after she graduated from medical school she went to volunteer medical service in africa wow, that's okay if communication was fine, but if there was no internet or telephone connection we could sometimes go two weeks without getting in touch with each other that often worries me now sell yourself in a minute, please i graduated from university two years ago and have since been working in an accounting company apart from the knowledge and skills related to my major i'm really good with computers i have a large amount of experience working with big data i think i'm well qualified for the accountant position here good, why did you choose this company? peterson has been my dream company all along you value progress and focus on the quality of work more than the quantity the flexible working hours and working from home is also attractive uh huh, what do you think is your biggest weakness well, when i work i often concentrate so much on what i'm doing that i may have a hard time dealing with things that come up unexpectedly that may be annoying to some people all right, thank you very much now sell yourself in a minute please i graduated from university two years ago and have since been working in an accounting company apart from the knowledge and skills related to my major i'm really good with computers i have a large amount of experience working with big data i think i'm well qualified for the accountant position here good why did you choose this company? peterson has been my dream company all along you value progress and focus on the quality of work more than the quantity the flexible working hours and working from home is also attractive uh huh, what do you think is your biggest weakness well when i work i often concentrate so much on what i'm doing that i may have a hard time dealing with things that come up unexpectedly that may be annoying to some people all right thank you very much so robert i guess you must be well into your third decade as a writer how is life treating you fourth decade as a writer hmm let me get my calculator out well, i guess i've been writing novels since the nineteen eighties so you're right but i have been writing all my life and to answer your question life has been treating me excellently can i ask you about your childhood were you a bookish child in a strange way i was a non reader until i turned nine i used to play basketball in the school yards of brooklyn my father was a high school teacher of physics and chemistry and he was a big reader he would bring books back from his high school library one day i read the last chapter of a book the title of which i've long since forgotten i liked it so much that i started reading the remainder interestingly i read the chapters backwards until i got to the beginning i really loved the experience and i soon became an eager reader then my father would bring piles of books, home from the library and i would read them all and they were all fiction i must add that i started reading the books from the start as opposed to working backwards so robert, i guess you must be well into your third decade as a writer how is life treating you fourth decade as a writer hmm, let me get my calculation later out well, i guess i've been writing novels since the 1980s so you're right but i have been writing all my life and to answer your question life has been treating me excellently can i ask you about your childhood were you a bookish child in a strange way i was a non reader until i turned nine i used to play basketball in the school yards of brooklyn, my father was a high school teacher of physics and chemistry and he was a big reader he would bring books back from his high school library one day, i read the last chapter of a book the title of which i've long since forgotten i liked it so much that i started reading the remainder interestingly i read the chapter's backwards until i got to the beginning i really loved the experience and i soon became an eager reader then my father would bring piles of books, home from the library and i would read them all and they were all fiction i must add that i started reading the books from the start as opposed to working backwards hello, i'm maxim vengarov and i'm very happy to share with you my connection with open chicafsky competition it was june nineteen eighty six when i came to moscow to participate in open tikovsky competition at the age of ten from cheleapinsk, this was actually the turning point of my life then i attended a world music festival in berlin, and it was interesting to say that a kid made his break through already at the age of ten, and that was me now as a judge of the competition i can see clearly the huge challenges facing the musicians in the competition where else can you see young players playing within ten days, classical works and russian traditional music before coming to the final stage as a musician it's important to give maximum attention and time to your own reflections of your own feelings so what i'm really looking for in the competition is to meet someone who can touch me musically i want to see the seed at least the seed of the great tree that can grow i would go for this violinist i would go for this musician hello, i'm maxim vengarov and i'm very happy to share with you my connection with open chicafsky competition it was june nine nineteen eighty six when i came to moscow to participate in open tykovsky competition at the age of ten from chelyapinsk, this was actually the turning point of my life then i attended a world music festival in berlin, and it was interesting to say that a kid made his break through already at the age of ten, and that was me now as a judge of the competition, i can see clearly the huge challenges facing the musicians in the competition, where else can you see young players playing within ten days, classical works and russian traditional music before coming to the final stage as a musician it's important to give maximum attention and time to your own reflections of your own feelings so what i'm really looking for in the competition is to meet someone who can touch me musically i want to see the seed at least the seed of the great tree that can grow i would go for this violinist i would go for this musician。 第二节到此结束。 could you please tell me how i can get to the student dorm building sure turn left when you walk out of this register office pass the bookstore and you'll see it we couldn't ask for a better day could we barbara, i know steve there isn't a cloud in the sky i love this time of year me too the flowers are beautiful but calling for rain though we got any plans this weekend we might start with our vegetable garden then the barbecue sounds fun but i still want to go fishing with george lisa, i saw jack this morning you mean the former designer of our company what's he doing these days? he said he's got a new job as an art reporter do you often go to the student union? andrew pretty much once in a while i meet friends there and i spend a lot of time organizing activities in different clubs or associations what do you think terry well, it has one more bedroom than the last flat and the sitting room is big but there is not enough cupboard space in the kitchen it's cheaper than the last one we saw and it is in a good neighborhood well, maybe we'll have to see if the agent has anything else to show us thank you missus johnson for what you've shared with our listeners now please tell us do you sometimes feel worried about your daughter now that she's grown up oh, i worry about her like all parents do after she graduated from medical school she went to volunteer medical service in africa wow, that's okay if communication was fine, but if there was no internet or telephone connection we could sometimes go two weeks without getting in touch with each other that often worries me now sell yourself in a minute please i graduated from university two years ago and have since been working in an accounting company apart from the knowledge and skills related to my major i'm really good with computers i have a large amount of experience working with big data i think i'm well qualified for the accountant position here good why did you choose this company? peterson has been my dream company all along you value progress and focus on the quality of work more than the quantity the flexible working hours and working from home is also attractive uh huh, what do you think is your biggest weakness well, when i work i often concentrate so much on what i'm doing that i may have a hard time dealing with things that come up unexpectedly that may be annoying to some people all right thank you very much so robert i guess you must be well into your third decade as a writer how is life treating you fourth decade as a writer hmm let me get my calculator out well, i guess i've been writing novels since the 1980s so you're right but i have been writing all my life and to answer your question life has been treating me excellently can i ask you about your childhood were you a bookish child in a strange way, i was a non reader until i turned nine i used to play basketball in the school yards of brooklyn, my father was a high school teacher of physics and chemistry and he was a big reader he would bring books back from his high school library one day, i read the last chapter of a book the title of which i've long since forgotten i liked it so much that i started reading the remainder interestingly i read the chapters backwards until i got to the beginning i really loved the experience and i soon became an eager reader then my father would bring piles of books home from the library and i would read them all and they were all fiction i must add that i started reading the books from the start as opposed to working backwards hello, i'm maxim vengarov and i'm very happy to share with you my connection with open tykovsky competition it was june nineteen eighty six when i came to moscow to participate in open tykovsky competition at the age of ten from chaleapinsk this was actually the turning point of my life then i attended a world music festival in berlin and it was interesting to say that a kid made his break through already at the age of ten and that was me now as a judge of the competition i can see clearly the huge challenges facing the musicians in the competition where else can you see young players playing within ten days, classical works and russian traditional music before coming to the final stage as a musician it's important to give maximum attention and time to your own reflections of your own feelings so what i'm really looking for in the competition is to meet someone who can touch me musically i want to see the seed at least the seed of the great tree that can grow i would go for this violinist, i would go for this musician。
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清华夫妇挑战今年高考数学题到底难不难?你当年高考数学考了多少分啊?我,没错,但是不知道哪里扣了我四分,考了一百四十六。这道题只有我一个人做出来,但还是被扣了四分。不,不懂,那你觉得今年的高考题难不难? 一般吧,那你给大家验证一下, ready go! 哎,现在这个脑子越来越没有之前那么好使了。大家都说今年计算量很大。是的,不过跟我们的年比还是有些差距,现在这完全小 k 反腐了。算一下这十七题, 这计算量确实挺大的啊。这只是第一步。我,我现在还不确定,能吗?反对,不过跟我们的年比还是有些差距。屋檐干净,脸有点疼。选择题一般是比较难的,现在我先看一下。第一, 你知 a 方加 b 方加 cc 方等于三。假设用方向函数, a 等于考乘以 r 法, c d 塔加考 c d 塔,这不就是小于等于根号六分就是考乘以 r 法就小于等于根号下六加三就等于根号,九就等于三。这个问题要证明, a 分之一加四分之一,这两个是不是有什么关系啊?一加二等于平方等于三等于平方。 之前有这个叫泰勒不等式是吧?对,现在有,我们当时是竞赛,采用竞赛用的方法都整上了,然后 a 加四, c 小于等于三的建筑,就是大于等于三的平方等于三等于三五得了,确实很难。

这样的文章内容不是很适合当高考题目,同学们都不一定读的特别懂,真的比全国已卷要难。亲爱的同学们,二零二二年全国假卷他来了, 量也不小啊。很多同学说甲劵比较难,今天徐老师来轻判一下 a 篇啊,主要靠你眼神好不好,直接一一找答案即可。一篇很短,但是有四道题,所以这文章肯定得读 b 篇文章。说实话,同学都不一定读的特别懂,但是选项还是可以比较好做的。四分钟过去了,就开始做 c 篇,你就发现全国甲券的试卷里好多这种俚语啊。文章是非常正宗的代刊里的文章啊。哎呀 天呐,可能要丢本文章,模模糊糊能看懂选项,感觉很像同学们,你不用方法和技巧怎么法题做?对呀!八分钟过去了,徐老师开始做地片, 就感觉文章很老很新鲜,没有科技感,跟你生活脱轨,好做吗?其实不好做你就不能多想,你想多了,他容易做不出来。 七选五很简单,是送分的,只要关键词都能一一对上,连词都没换,满心填空,我还是先看一遍。呃, 哇塞,这篇文章很适合女生做呀,海底是有点感情经历的,所以看完文章你才知道两个人是慢热,后面才真的在一起,那刚开始交往的时候是有所保留的, 你说我们的高三的学生怎么会呢?这样的文章内容不是很适合当高考题目,不管咋说,我先给他评论。语法填空很简单很舒服,应该不会丢分。我觉得咱们全国讲究的文章选的不太好,然后跟孩子们平常接触的匹配不上,文章选的有点偏了,幸亏选 选项出的还好,但是同学们想得一个高分还是非常有难度的。失败的一份试卷,我好气啊,徐老师花了二十二分钟写完了全国假证回来对下答案,现在是大搜查时间, 阅读虽然处的不好,但我觉得阅读吧,我应该不会错。完形填空错了两道一点五三分,语法填 都对的,抛掉作文和听力还有改错,一共是八十五分。徐老师的成绩是八十二分,错了两道完形填空,我还挺不服的,这试卷让我觉得真的比全国已卷硬盘。巴山夜雨涨秋池。

二零二三年的全国奖卷是让我最期待的一份试卷,同学们都说难,哭天喊地的,我太想做这份试卷了。第一题也是一道旅游题,第一题讲两个餐厅的共同点,我建议同学们先去看那个段落上,先取一小时。第一篇还是比较简单,一篇地道推测,词语不好做, 这几个选项意思太接近了,尤其是 a 和 c, b 片。第三题有坑啊,很多同学不认识 deposit 这个词,是存款还有押金的意思 不好做。完了,这个碎片太难了,里头太多的人名,太多的书名,都给我整乱了,我现在得老老看着什么字母,我就怕给他张管理再把题做错了,特别费脑子。 碎片看着不难, tz 恶心,碎片不要,这点方法和技巧你都不知道这篇文章在说啥,晕,太呀太晕了! if we have broke up 一篇上来第一段就感觉没有新高考一卷好落,虽然文章没有什么深度,但到底在说啥,主旨是什么?这阅读理解出的不好,出卷老师觉得真简单,同学们觉得是真难受,我懂 more the city is the things you said if we ever broke up。 七千五还比较友好。虽然有一定难度, 环境填空出的很好,但是这个难度比新高考要高出好几个度来。哎呀,坐着有点晕晕的,天呐,最后一篇语法填空式讲预言也是不接地气。 经过了四十分钟,薛老师做完了全国奖卷,下面来对一下答案,累了给整难受了。薛老师错了,一道完形填空,一道语法填空。科目题讲语法填, 一共七十五分,徐老师错了,一个完形,一个语法填空,扣掉三分,最后是七十二分。全国假卷给我的感觉就是文章很虚,很哲学,很松狂。做完这个试卷,我明白同学们的感受了,确实难,真的不好做。这试卷比新高考一卷难做多了。

看二零二三年全国假卷最难的选择题,这个题怎么办?其实很简单,我们看拿到一个椭圆,首先判断 x 型还是 y 型, 这一定是 x 型,因为 x 方下面大,然后写出 a 方和 b 方,当然你也可以不用写减,因为是选择题, a 方等于几啊?等于九,所以 a 等于三, b 方等于六, b 等于根号六, c 方等于 a 方,减 b 方 c 等于根号三。然后画图 坠,把图稍微画的大一点。 x 型椭圆稍微画的大一点,不用太标准。 那我们看 f 一, f 二为左右焦点,这是 f 一,这是 f 二, p 点为椭圆上一点,我们一般点在第一象限 cruising 角 f e p f 二。好,我们知道这个角度 求 o p, 这是 o 点。其实这道题有很多的解法, 我们用我们平时教的最常规最常规最常规的方法来讲,移动点到两定点,距离之和等于长处二 a, 我的习惯是设边长为 m n, 当然了, f e f 二是焦距二 c 二倍根号三,这是我们名门学府张超带你学高中数学这本书里面讲的最基本的方法,不要用套路 来。移动点到两定点,距离之和等于长数二 a, 那 m 加 n 不就等于二 a 吗? 当然二 a 就是六,有两焦点用,定义有直角 购物定理不是直角用余弦定理。很简单,我肯定要用余弦定理。所以零边的平方和 减对边的平方二得四三 c 十二除以二倍,零边就等于五分之三。 一是怎么化减?讲了一万遍了,平方和二是搭配 c, m 方加 n 方加二 m n 等于三十六,那不就是三十六减二 m n 吗? 二是区分母,你看 m 方加 n 方减去十 十二等于五分之六倍 m n。 好,它俩相等怎么办?很简单,三十六减二, m n 等于十二,加上五分之六倍 m n, 你走三十六减十二等于几啊?二十四,这是五分之十,那不就是五分之十六吗? 你不用想了,他们都有八来,同时乘以八分之一,三五分之二 m n, 那你看 m n 出来没有? 三五一十五,二分之十五。那既然 m n 出来了,那 m 方加 n 方不也出来了吗?你看等于几? 二分之十五乘以二十五,三十五减十五等于二十一。这就是最基本的化解一个最基础的考题。一天到晚抱怨难难难,我说让你把每个题目做三遍,你为什么不听呢? 来求 o p, 我们看 o p 怎么办?其实你就不用想 o 是什么点,终点,那这个 o p 不就是中线吗?所以这个 也很容易用我们书上讲的中线定理来求中线的长度。 怎么办?你看,不要跑,你要为全国的学生服务,躲起来干嘛?好书就不能掩盖,你看 o 为 f 一 f 二的终点 不做了。那你看 p u 项链等于什么?不就等于二分之一 p f 一项链加 p f 二项链吗? 项链的平方等于项链的魔方,一平方不就出魔了吗?不就出长度了吗?所以 p o 的平方就是 p o 的魔方,等于四分之一倍 p f 一的平方就是 p f 一的魔方,加 p f 二的平方就是 p f 二的魔方。 加两倍的 p f 一乘 p f 二, p f 一向量乘 p f 二向量向量相乘等于膜膜 q n 加角,那不是 p f 一的膜 p f 二膜 q n 加角吗? q n 加角五分之三。 来,我们看,四分之一不变 m 方加音方我刚刚已经求过了二十一,这都顺理成章的事情。二倍 m n 就是十五, 十五乘以三四十五,四十五除以五九,所以就等于四分之二三十。 人家求 o p 一定不要忘了开方,所以 o p 的膜等于几啊?等于二分之二三十。 当你一天到晚抱怨难的时候,你为什么不能脚踏实地的去努力?高考只有二十二个题, 抓住二十二个题的模板,做透做熟,数学变得很简单。

环形填空第一句话同学们看完了之后根本不知道他在说什么,就这种题,错过两三道四五道都很正常。今天徐老师和大家一起来做二零二一年全国假卷,按说应该是全国最简单的一份,那么咱们还是来看一下,事实果真如此吗? 从这个篇幅上来看,都是比较短,然后词汇看上去没有特别难的词汇, 其中第二套题真的得细心的看一看,排一排,要不很容易选错 第一篇,第二篇都是书铭文,但是吧,嗯,特别考你的眼神,其中有一个动物的名字,我找了好久才找到。 做完了三篇阅读,徐老师感受到的是推理题的难度,然后从 a 篇、 b 篇、 c 篇开始,每篇都有,我认为这个试卷出的还真是挺好的。接下来咱们再看地篇, 地篇也是一样啊,上来两道题全是推理题, 第一篇的思路非常的清晰,十三题是一个推理判断,十五呢是阻止大意,这两道题很容易失分,但是十三题的这个 c 选项你好好看一下,他和原文章其实是通话了,概念的十五题,大家可以通过排出法来做全国假卷的这个阅读理解给我特别浓的感受是 细节题很简单,直接找全能找到。推理判断题出的很柔和,但是每一个阅读理解中都有很多同学在这地方 容易失分阻止,大一题还好,就考了一道题。所以说你的主维的努力方向就是应该把推理判断题做好, 七选五啊,同学们一定要看好词汇啊,词的意思是不一样的。这篇文章讲的是交换服装,不是去捐赠服装,所以有一道题太有干扰性了,就是易选项七选五,还是挺容易出错的。 我相信这个完形填空第一句话同学们看完了之后根本不知道他在说什么,是一份道歉信。这种题不是很好做,因为有一个特别长的场景描写,就这种题错个两三到四五道都很正常。 这个完形我觉 绝对比我们新高考依旧的晚行都难。真的,同学们太容易丢分了,凭了吗我, 哦,没有屏哎,六个地!徐老师,我真的也犹豫了,但我答案不改了啊,哇,难度好大呀! 这次的语法填空比去年就简单多了啊,今年的好得分太好得分了,顶多错一道吧。徐老师花了二十五分钟写完了全国假卷,下面来对一下答案。 徐老师坚定立场,完形没有追求配平,去掉作文和听力,满分是八十五分,徐老师的成绩也是八十五分。这个试卷给我的感受就是绵绵中透着刺,所以咱 咱们同学一不小心就会被刺伤。整体来说,咱们同学一百三以上很好达到,但是我想一百四以上的同学真的不多,其他省事同学们也可以来练一下全国假卷,尤其是我们高二的同学们。

大家好,今天我来讲一下二零二三年全国假卷的立体几何大题。通过这道大题的讲解,我其实想告诉大家一个道理啊,就是间隙法在立体几何大题中真的不是万能的,很多立体几何的大题,其实不能用间隙法或者用间隙法,它的计算量太大 来,真正聪明的同学啊,应该是立体几何,他的间隙的方法还有纯几何的方法都应该掌握,都应该灵活运用。我们看一下这道题,他说的是三棱柱中,这个三棱柱的话,他可不是直三棱柱,他这个侧棱是斜着的啊,这一看就能看出来, 然后呢,继续往后读,他说什么呢?他说的是 a a 一等于二,就是侧棱长啊, a a 一, b b 一, c c 一其实都等于二。他告诉你谁垂直于底面呢?他说的是 a、 e、 c 这样一个侧面的面对角线, 它是垂直于 a、 b、 c 这样一个底面的,它垂直啊,并且告诉你这个角,也就是说 a c 和 b c 夹角是九十度吧,也就是说什么呢? a e、 c a c、 b c 这三条线两两之间是互相垂直的,可能到这之后的话,很多人就直接 x 轴、 y 轴、 z 轴写上了。这第一问肯定不是这样来正的啊,根本就没有必要去间隙,不是用间隙法来做的。那么我们看一下 它最后还有一个条件, a 一到平面,它的距离是一。我呢,反正读完题之后,我就看到了一个一,一个二, 反正再加上两个垂直呗。好,一二垂直,那你能不能想到在直角三角形中,斜边中线等于斜边一半这样一个一比二的道理啊?肯定能吧,那么我们来看一下 这道题啊,第一问的话,真的特别简单,我们就找重点啊,斜边重点等于斜边一半,哪个平面呢?我们把这个侧面啊,也就是 a e a c c e 这样一个平面呢,他这个侧面画出来他的平面图,画完这个平面之后的话,首先垂直底面,不用多说了吧, 既然你这个 a e c 是垂直于整个底面 a b c 的,它肯定垂直于 a c 啊,那进而也垂直于 a e c e 啊,是不是?好了,肯定这个垂直是好得出来的,得完这个垂直之后的话,为什么要找终点呢?首先斜边中点的一些密码,这是二吧, 这是一吧,点 o 在哪啊?你要注意啊,在这样一个立体几何的原图中,这个点 o 它指的是 c c 一的终点,那么点 o 它记在 c c 一上面,那肯定就是在平面 b c c b 以上的。好了,不用多说了,看好了啊,因为什么我们分析一下这道题,很容易证明呢?这个 a e c 是垂直于 a e c e 的, 并且呢,再来看这个 a e o 斜边中线等于斜边一半吧,它就是斜边 c c e 的一半,等于一啊, 那么既然点 o, 他是在什么?他是在平面上的,我知道了,其实你看了啊,因为 a 一点到,请告诉我,在初中阶段学过这样一个定理,没有点到直线,除以现在最短吧,那高中的话,我们就知道了,点到什么? 点到平面,他其实也是垂线段最短的,最短最短,最,就是唯一的意思啊,就是那个垂足只有一个的意思, 其实那个点 o, 它就是点 a 一到平面,哪个平面我写清楚啊?到平面 b c c 一 b 一的这样一个垂组,那个点 o 就是垂组,懂了吧?它的距离 我写清楚了,距离为一,所以再加上什么,所以我就知道了呀,其实 aeo 它就是垂直于整个平面 b c c 一 b 一的,剩下不用多说了吧。然后呢,也很容易证明的一点是什么?也很容易证明的一点是,首先我们可以通过证明谁通过证明这个 b c 垂直于整个 侧面 a c c 一 a 一,然后进而证明两个平面,也就是说平面 a c c 一 a 一,它是垂直于我们的 目标平面 bccebe 的。为什么呢?原因很简单,首先我已经分析过了,两两之间互相垂直吧,既然他这样一个 bc 垂直于平面里头两条相交直线,所以就垂直于整个平面,进而呢,就得出来这样一个面面垂直了,那得出来面面垂直的话,你说这两个平面 它的胶线是谁啊?这肉眼可见的嘛。胶线,两个垂直,平面的胶线就是它垂直于胶线则垂直,平面,垂直于平面则垂直于它俩的胶线吧,这是一个定力点,所以其实 a u 它是垂直于整个 c c e 这样一个胶线的。 那既然证明完这一条的话,所以不用多说了吧。点 o 既是垂直的,它也是终点,所以 a u 是什么?是中垂线呀?中垂线它的性质定理是什么?中垂线上的点到 线段,两个端点距离相等,所以它这个 a e c 等于什么?就等于 a e c e 啊,那 a e c e 显然是等于 a c 的,不就证明完了,就这么简单, 懂了吧?点到平面垂线呢,最大它的距离就是一,这道题给的太明显了,一个二,一个一,加上两个垂直,肯定是要想直角三角形中斜边中线等于斜边一半这样一个定律的。我们继续来看第二问啊。第二问的话,如果你要选择用间隙的方法,不是说不能做,我说一下 间隙的方法有什么缺点啊?间隙的方法的话,咱们根据第一问这个图形的话,很容易判断这个 a e c 和 a c 它的距离的话,肯定是等于根号二的啊,这个都很容易算,也就是说 a 一点,还有 a 点我们坐标呢,都是可以写出来的, 然后包括这个 a 一点,我们可以写什么?那 z 轴当然是零,逗号零,根号二了,但是 你想写 b 点吧, b 点的坐标你得假设了,只能假设零多少 t 零,你还得把这个 t 求出来。根据线与线的这样一个距离, 是不是还得做辅助线,比如说 b 点啊,做了一个垂直,比如说垂直于他,然后 bh 的长度是等于二的,然后再进而用什么用 啊?要么是点臂到向量的距离公式,要么就是根据平面向量他这样一个投影的公式,当然 他这个空间下来也可以用投影公式啊。所以说这道题的话,剩下的计算量会很大,然后再把这个 t 具体的话是等于根号三算出来。这个咱不推荐这种方法,你看看如果这道题 用纯几何的方法,他有多简单啊。看了,首先这道题的话,有一点很明确,你看刚才我们在证明的时候,这个 aeo 是不是垂直于整个平面点好, 这道题的话,我连接一下 a b 一,为什么连接这个 a b 一呢?道理很简单,因为你求的就是 a b 一和整个目标平面的它夹角的正线值。这个平面的话不用多说了吧,我就擦掉了啊,平面是在这样一个位置的,在后边, 那么看线面角,它的平面角需要找出来。其实线面角它得有三个要素。首先第一个要素就是 就是斜足,斜足就指的是斜线 a b 一和平面的焦点,斜足当然是 b 点了,不用多说,斜点的话是我发明的这样一个名词,其实就是斜线上的异于斜足的一个点,那就写成 a 点, 没他有这样一个点过, a 点还得做出些什么,他其实缺少了。第三个要素,缺少了垂足,对吧?比如说我们做 am 是垂直于整个平面哦,所以当你想到这一点的时候,我觉得辅助线其实就很明确了。首先第一条辅助线我先告诉你啊,我们只需要做 am 怎么样? am 是平行于 d 问中的 aeo 的,他俩之间是平行的呀,再加上别的平行交臂啊。知道了,那有了这样一个平行之后的话,再加上这个 aae 是平行于 cce 的,肯定很容易出现一个平行字边形吧。咱们写上啊, 这个平行四边形呢,就是 a m o a e 能得出来,那根据平行四边形结论的话,是不是很容易就得出来 a m 的长度了呀?它和 a u 的长度其实都是等于一的, 对不对?那做完这个之后的话,效果是什么?你为什么要做这个 a m 平行于它呢?道理很简单,因为在第一文中我们已经说明了 a e o, 它就是 下面垂直的那样一条垂线,它是垂直于整个目标平面 b c c 一 b 一的。所以既然平行的话,你这个 a m 是不是也垂直于整个平面 b c c 一 b 一啊?它想告诉你的就是说 m 点, 他就是线面垂直的这样一个垂足呀。得出这个垂足来之后,你看线面角的三要素都有了啊, a m 是整个平面 b c c e b 一他的垂足吧,点臂是斜足,他是,哦,连接起来我知道了,如果我再说的明 这些,其实就是这么的回事,这是 a 点,这是 m 点,这是 b 一点,那这个 am 等于一。已经算出来了啊,它的长度等于一,那么其实我们只需要算一下 bm 或者说啊 ab 的长度就行了。 这道题我们选择算一下 b、 e、 m 的长度,是不是就把这个正弦值算出来了呀?因为这道题,他这个线面角的平面角其实就是垂啊,我写出来他这个线面角的平面角,其实就是让我们求哪个角,求这个角 a b e m 它的正弦值,显然它的正弦值 a b e m 是等于谁比上谁了,你就告诉我,其实就是等于刚才我画的那样一个直角三角形 a m b e 中的它这样一个 a m 的长度,再比上谁的长度,再比上 a、 b、 e 的长度, a b e 长度的话,我觉得不太好算啊, a b 是斜边,我们就用勾股定理来算吧, a m 显然等于一, 它下边这个斜边。关于这个 a b 一的话,我们就用这个 a m 的长度的平方,再加上 b e m 的平方。所以这道题其实我们只需要算,是到现在为止我分析清楚了,只需要算 b e m 的长度啊,你既然要算 b e m 的长度的话,是不是要涉及到勾股定理啊? b e m。 好,现在看了 b e m 可以放,把它放到哪个直角三角形中看了, 这是不是一个直角三角形?我们分析一下,肯定是直角三角形的吗?其实后边的话,他这个平面你看看好了啊, bc 是不是垂直的?是啊,垂直于 cce 啊,所以后边这样一个平面,他其实是一个矩形的, 对吧?这个很容易得嘛,根据九十度,根据垂直就可以得出来矩形。矩形的话来看,是不是需要先求一下 b 一 c 一的长度,再求一下下边?我觉得不用多求了吧,肯定很容易求出来的啊。我们容易得到的是, 其实有一个全等的,谁呢?三角形 a m c, 它是全等于三角形 a e o c e 的,你这个 c e o 的长度等于一点二,是终点吗?它的长度也是一,所以下面这个 c m 的长度, 它是不是就等于这个 c e o 的长度等于一?它都是一啊,所以我这个 c e m 的长度就等于一加一加一等于三就算出来了。关键在于我这个 b 一 c 一是多长? b 一 c 一。好,第二条辅助线看好了啊,第二条辅助线要干嘛呀?要找一个 b e c e 的终点,因为点 o 已经是终点了,点 n 也也是终点那 o n 的长度,哎,能理解吧。看了啊,千万不要着急,这道题首先看好了。 嗯哪呢,我要换一种颜色吧,千万不要混了看。首先, c c e 它是怎么样的?它是垂直于 a e o 的吧。第一问已经解决过了, 并且呢,根据矩形的特性,这个 c c e, 它也是垂直于这个 o n 的吧?你垂直 c c e, 垂直于整个平面里头两条相照直线,它不就垂直于整个平面 a e、 o n 吗? 所以进而就推出来,这个 c c、 e 是垂直于整个平面里头的最后一条线。垂直于谁?垂直于 a e n? 当你求完垂直于 a e n 之后的话,你这个 a、 e、 n 是不是垂直于 b b e? 来看看 看啊,新增那些条件,括号二,增加了个什么条件?他说这个 aae 和 bb 这两条平行线之间距离就是二,那不就是点 ae 到 bbe 的距离,这个 aen 不就是那个距离等于二吗?对吧?根据已知条件,这个 a、 e、 n 的长度,根据垂直,他就是两个平行线 a a 一和 b b 的距离等于二。好了,剩下呢,不用多说了吧,所以能算了啊。好,我把剩下的东西呢,该擦的都擦掉来。 a e o 等于多少吧?等于一吧。然后呢,这个 a e n 等于多少?等于二吧。那你说这个 o n 等于多长呢? o n 肯定是等于二的平方减去一的平方。勾股定理啊,它是等于根号三的, o n 等于根号三,所以说 b 一 c 一其实也是根号三,所以 记得很清楚啦, b e m, 它的长度是等于根号下 b e c e 的平方,再加上 c e m 的平方, b e c e 的话,平方就是三,这个 c e、 m 的平方就是九,对吧?三的平方等于九,等于根号十二二倍根号三呀。 那么我们要算的最终的结论不是就有了吗?刚才在这样一个位置我已经写过了这个三角 a b e m, 它是等于谁比上谁的,它就是等于这样一个直角边 a m 比上斜边 a b 一的其实也好算啊。那 a m 的话就是一它的平方啊。根据勾股定理,这个 a b 一,它是等于 a m 的平方,其实就是一的平方,再加上谁再加上这个 b e m 的平方吧。那显示等 等于十二点,最终结果就是根号下一加十二,根号十三分之一,也就是说十三分之根号十三就解决完了。所以这道题的话,其实我主推的方法是什么?就是几何的方法。当然,如果你真的要用间隙的方法也能算啊。也不是说不能算,那么真正 聪明的同学是立体几何,间隙的方法也得掌握。还有立体几何纯几何的方法也得掌握清楚了吧。分享课堂知识,感受数学之美。我是安分老师,下节课再见!