{"id":327,"date":"2012-03-01T01:31:33","date_gmt":"2012-02-29T16:31:33","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=327"},"modified":"2021-11-04T14:48:34","modified_gmt":"2021-11-04T05:48:34","slug":"htb200804","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/htb200804\/","title":{"rendered":"\u4e00\u6a4b\u59272008\uff1a\u7b2c4\u554f"},"content":{"rendered":"
\n

\u6b63\u56db\u9762\u4f53 OABC \u306e \\(1\\) \u8fba\u306e\u9577\u3055\u304c \\(1\\) \u3068\u3059\u308b.\r\n\u8fba OA \u3092 \\(2:1\\) \u306b\u5185\u5206\u3059\u308b\u70b9 P , \u8fba OB \u3092 \\(1 : 2\\) \u306b\u5185\u5206\u3059\u308b\u70b9\u3092 Q \u3068\u3057, \\(0 \\lt t \\lt 1\\) \u3092\u307f\u305f\u3059 \\(t\\) \u306b\u5bfe\u3057\u3066, \u8fba OC \u3092 \\(t : 1-t\\) \u306b\u5185\u5206\u3059\u308b\u70b9\u3092 R \u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000PQ \u306e\u9577\u3055\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u25b3PQR \u306e\u9762\u7a4d\u304c\u6700\u5c0f\u306b\u306a\u308b\u3068\u304d\u306e \\(t\\) \u306e\u5024\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n

    (1)<\/strong><\/p>\r\n

    \\(\\overrightarrow{\\text{OA}} =\\overrightarrow{a}\\) , \\(\\overrightarrow{\\text{OB}} =\\overrightarrow{b}\\) , \\(\\overrightarrow{\\text{OC}} =\\overrightarrow{c}\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\n\\left| \\overrightarrow{a} \\right| & = \\left| \\overrightarrow{b} \\right| = \\left| \\overrightarrow{c} \\right| = 1 , \\\\\r\n\\overrightarrow{a} \\cdot \\overrightarrow{b} & = \\overrightarrow{b} \\cdot \\overrightarrow{c} = \\overrightarrow{c} \\cdot \\overrightarrow{a} = 1 \\cdot 1 \\cdot \\cos 60^{\\circ} = \\dfrac{1}{2} , \\\\\r\n\\overrightarrow{\\text{OP}} & = \\dfrac{2}{3} \\overrightarrow{a} , \\ \\overrightarrow{\\text{OQ}} = \\dfrac{1}{3} \\overrightarrow{b} , \\ \\overrightarrow{\\text{OR}} = t \\overrightarrow{c}\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\n\\left| \\overrightarrow{\\text{PQ}} \\right|^2 & = \\left| \\dfrac{2}{3} \\overrightarrow{a} -\\dfrac{1}{3} \\overrightarrow{b} \\right|^2 \\\\\r\n& = \\dfrac{4 -4 \\cdot \\frac{1}{2} +1}{9} =\\dfrac{1}{3}\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n\\text{PQ} = \\underline{\\dfrac{\\sqrt{3}}{3}}\r\n\\]\r\n

    (2)<\/strong><\/p>\r\n

    \\[\\begin{align}\r\n\\left| \\overrightarrow{\\text{PR}} \\right|^2 & = \\left| \\dfrac{2}{3} \\overrightarrow{a} -t \\overrightarrow{c} \\right|^2 = \\dfrac{4}{9} -\\dfrac{2t}{3} +t^2 , \\\\\r\n\\overrightarrow{\\text{PQ}} \\cdot \\overrightarrow{\\text{PR}} & = \\left( \\dfrac{2}{3} \\overrightarrow{a} -\\dfrac{1}{3} \\overrightarrow{b} \\right) \\cdot \\left( \\dfrac{2}{3} \\overrightarrow{a} -t \\overrightarrow{c} \\right) \\\\\r\n& = \\dfrac{4}{9} -\\dfrac{t}{3} -\\dfrac{1}{9} +\\dfrac{t}{6} \\\\\r\n& = \\dfrac{1}{3} -\\dfrac{t}{6}\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \u25b3PQR \u306e\u9762\u7a4d \\(S\\) \u306f\r\n\\[\\begin{align}\r\nS & = \\dfrac{1}{2} \\sqrt{\\left| \\overrightarrow{\\text{PQ}} \\right|^2 \\left| \\overrightarrow{\\text{PR}} \\right|^2 -\\left( \\overrightarrow{\\text{PQ}} \\cdot \\overrightarrow{\\text{PR}} \\right)^2} \\\\\r\n& = \\dfrac{1}{2} \\sqrt{\\dfrac{1}{3} \\left( \\dfrac{4}{9} -\\dfrac{2t}{3} +t^2 \\right) -\\left( \\dfrac{1}{3} -\\dfrac{t}{6} \\right)^2} \\\\\r\n& = \\dfrac{1}{2} \\sqrt{\\dfrac{11 t^2}{36} -\\dfrac{t}{9} +\\dfrac{1}{27}} \\\\\r\n& = \\dfrac{1}{2} \\sqrt{\\dfrac{11}{36} \\left( t -\\dfrac{2}{11} \\right)^2 +\\dfrac{1}{27} -\\dfrac{1}{99}}\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \\(S\\) \u3092\u6700\u5c0f\u306b\u3059\u308b \\(t\\) \u306e\u5024\u306f\r\n\\[\r\nt = \\underline{\\dfrac{2}{11}}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u6b63\u56db\u9762\u4f53 OABC \u306e \\(1\\) \u8fba\u306e\u9577\u3055\u304c \\(1\\) \u3068\u3059\u308b. \u8fba OA \u3092 \\(2:1\\) \u306b\u5185\u5206\u3059\u308b\u70b9 P , \u8fba OB \u3092 \\(1 : 2\\) \u306b\u5185\u5206\u3059\u308b\u70b9\u3092 Q \u3068\u3057, \\(0 \\lt t \\lt 1\\) … \u7d9a\u304d\u3092\u8aad\u3080 →<\/span><\/a>","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[57],"tags":[146,16],"class_list":["post-327","post","type-post","status-publish","format-standard","hentry","category-hitotsubashi_2008","tag-hitotsubashi","tag-16"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/327","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/comments?post=327"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/327\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=327"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=327"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=327"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}