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数学 高校生

なぜOG:GH=1:2なのですか?

考え方 練習 348 例題 348 オイラー線 △ABCの外接円の中心を0とし、頂点A,B,Cの点Oを基点とする 位置ベクトルを,それぞれ a, , こ とする. 位置ベクトル h =a+b+c で表される点をH, △ABCの重心をGとするとき,次の 問いに答えよ. $JCA (1) 3点 0, G, H は一直線上にあることを示せ . (2) 点Hは△ABC の垂心であることを示せ . SONS (1) 3点O,G,Hが一直線上にある OH =kOG の形で表せる (2)点Hは△ABCの垂心 Focus また、点は外接円の中心だから |==|| 3.685206(OA+OB+OC)-OGR FOR =3OG-OG=20G AHBC, BHICA つまり, AH・BC=0, BH・CA=0 つまりよって,3点0,G,Hは一直線上にある. (別解) GH = AH-AG=OB+OC- (OG-OA) の大温kg ADCƏ (1) OH=a+b+c, OG=1/(1+6+2) より, OH=3OGOH=kOG の形で 3 よって、3点0, G, Hは一直線上にある . ができる (2) 点Oは△ABCの外心だから, |a|=|8|=||| AH・BC=(OB+OC) ・(OC-OB) =(c+b). (c−b) >5508 よって, BH CA=(OA+OČ) (OA-OC)B ^¹ =(a+c)·(a−ĉ)¯AS 12-10 AH•BC=0\ 0803 H = 0 を利用 (内積) 5 3 ベクトルと図形 61 ** A O G 線分が垂直 注 三角形の外心O, 重心G,垂心Hは一直線上にあり, OG: GH = 1:2 である. (直線OGH をオイラー線という.) M C OG: GH=1:2 AH-OH-OA, OH = OA+OB+OC より 08055-3-57 (0) 0200315 20 AH=OB+OC OĞ=(a+b+c) =lap²-1c²²=005 (SCE BH・CA = 0 よって, 以上より, AH⊥BC, BHICA だから,点Hは△ABC A = 0, BH ±0 とし ても一般性を失わない. の垂心である. BH=OH-OB OH = OA+OB+OC より, BH = OA+OC rernzelni. の方面 例題 348 において, 点Cを通り外接円の直径となるようなもう一方の円周上 の点をEとするとき,四角形 AEBH は平行四辺形となることを示せ. →p. 63028

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英語 高校生

4の解き方がわかりません。 例えば(a)で、続く文であくびを謝っていることも全て把握した上で、挨拶なのでpolitelyかと考えたのですが、なぜ誤りなのか教えていただきたいです。

13 Lucy is a British schoolgirl who lives in Oxford. She is talking to her father, Fred, in the kitchen of their home. Read the conversation below and answer the following questions. Lucy: (a) Good morning, Dad. Oh, sorry for yawning. What are you doing with that microwave oven? It looks heavy. Do you need a hand with moving it? Fred: I think I'll be OK, Lucy. I'm just going to put it in the car and take it to the city dump. Lucy: Couldn't you send it away to be repaired? Fred: It's ten years old and well out of warranty now, so I very much doubt that the manufacturer would do it. They probably don't even carry the spare parts anymore. Lucy: That's a shame. Oh, I know what! Why not take it to the Repair Café near my school? Fred: What's that? I've never heard of it. Lucy: It's brilliant! We visited it as part of our environmental science course recently. It's a meeting place where people can get together to mend broken items cooperatively. And have a chat and a cup of coffee! Fred: I'm all ears. Tell me more. Lucy: Well, the first Repair Café was started by a Dutch woman called Martine Postma in Amsterdam in 2009. (1) 彼女は,使い捨て文化で環境が破壊されて, ゴミの量が地球規模で増えることを心配してたん . She wanted to find a local solution to this global problem. Fred: That's what they call "thinking globally, acting locally," isn't it? Lucy: Exactly. She also wanted to address the decline in community spirit amongst urban dwellers and do something about people's loss of practical skills and ingenuity. Fred: And the idea (2) caught on? Lucy: Very much so. The concept has grown into a global movement. The one in Oxford started about four years ago. Fred: Impressive! What kind of things do they repair? Lucy: Oh, all sorts. Electrical appliances, clothes, furniture, crockery, bicycles, and even toys. Fred: How successful are they at repairing things? Lucy: Well, of course, they cannot guarantee to fix every item brought to them, but they have a fairly good success rate. One study found that on average 60 to 70 percent of items were repaired. The rate is higher for some items such as bicycles and clothes but lower for things like laptop computers. Fred: I can understand that. Just between you and me, I think some electronics manufacturers deliberately make products in such a way that you cannot disassemble them and repair them unless you have specialized tools and equipment. Lucy: Yes, and that's where Repair Cafés can help. But these cafés are not just about repairing things for people. They are places where we can meet others, share ideas, and be inspired. The volunteer repairers are very keen to involve the visitors in thinking about the repair and actually carrying out the repair themselves. They also encourage people to think about living together in more sustainable communities. Fred: I suppose you could say they are about repairing our minds, not just our things. Lucy: Quite so, although often the two are very closely related. Many people attach (3)sentimental value to old things that might, for example, be part of their family history. Fred: I see what you mean. It almost sounds too good to be true. (a)Is there a catch? For example, how much does it cost? Lucy: Advice and help from the repairers is free, but people who use the café are invited to make a donation. That money is used to cover the costs involved in running the café. If specific spare parts are needed, the repairers will advise you on how to obtain them. Fred: Well, that's marvelous! (e)Then I'll take this old microwave there. Are they open today? Lucy: Yes, and I'll come with you. I've got a pair of jeans that are badly in need of some attention. QUESTIONS 1. Translate the underlined part after (1) into English. 2. What does the underlined phrase after (2) mean? Select the most appropriate expression from the list below. (A) became popular (B) hit a dead end (C) occurred to you (D) played a significant role (E) worked in practice 3. The underlined phrase after (3) means the value of an object which is derived from personal or emotional association rather than its material worth. Give ONE object that has "sentimental value" for you and explain why it has such value. Your answer should be between 15 and 20 English words in length. (Indicate the number of words you have written at the end of your answer.)

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