{"id":1243,"date":"2015-08-22T20:13:58","date_gmt":"2015-08-22T11:13:58","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1243"},"modified":"2021-09-23T22:51:03","modified_gmt":"2021-09-23T13:51:03","slug":"tok201502","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tok201502\/","title":{"rendered":"\u6771\u5de5\u59272015\uff1a\u7b2c2\u554f"},"content":{"rendered":"
\n

\u56db\u9762\u4f53 OABC \u306b\u304a\u3044\u3066, \\(\\text{OA} = \\text{OB} = \\text{OC} = \\text{BC} = 1\\) , \\(\\text{AB} = \\text{AC} = x\\) \u3068\u3059\u308b.\r\n\u9802\u70b9 O \u304b\u3089\u5e73\u9762 ABC \u306b\u5782\u7dda\u3092\u4e0b\u308d\u3057, \u5e73\u9762 ABC \u3068\u306e\u4ea4\u70b9\u3092 H \u3068\u3059\u308b.\r\n\u9802\u70b9 A \u304b\u3089\u5e73\u9762 OBC \u306b\u5782\u7dda\u3092\u4e0b\u308d\u3057, \u5e73\u9762 OBC \u3068\u306e\u4ea4\u70b9\u3092 H' \u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(\\overrightarrow{\\text{OA}} = \\overrightarrow{a}\\) , \\(\\overrightarrow{\\text{OB}} = \\overrightarrow{b}\\) , \\(\\overrightarrow{\\text{OC}} = \\overrightarrow{c}\\) \u3068\u3057, \\(\\overrightarrow{\\text{OH}} = p \\overrightarrow{a} +q \\overrightarrow{b} +r \\overrightarrow{c}\\) , \\(\\overrightarrow{\\text{OH'}} = s \\overrightarrow{b} +t \\overrightarrow{c}\\) \u3068\u8868\u3059. \u3053\u306e\u3068\u304d, \\(p , q , r\\) \u304a\u3088\u3073 \\(s , t\\) \u3092 \\(x\\) \u306e\u5f0f\u3067\u8868\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u56db\u9762\u4f53 OABC \u306e\u4f53\u7a4d \\(V\\) \u3092 \\(x\\) \u3067\u8868\u305b. \u307e\u305f, \\(x\\) \u304c\u5909\u5316\u3059\u308b\u3068\u304d\u306e \\(V\\) \u306e\u6700\u5927\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

    BC \u306e\u4e2d\u70b9\u3092 M \u3068\u304a\u304f\u3068, \u5bfe\u79f0\u6027\u304b\u3089, H , H' \u306f\u5e73\u9762 OAM \u4e0a\u306b\u5b58\u5728\u3059\u308b.
    \r\n\\(\\overrightarrow{\\text{OM}} = \\overrightarrow{m} = \\dfrac{\\overrightarrow{b} +\\overrightarrow{c}}{2}\\) \u3068\u304a\u304f.
    \r\n\u25b3OAM \u306b\u7740\u76ee\u3059\u308c\u3070\r\n\\[\r\n\\text{OM} = \\dfrac{\\sqrt{3}}{2} , \\ \\text{AM} = \\sqrt{x^2 -\\dfrac{1}{4}}\r\n\\]\r\n\u306a\u306e\u3067, \u4f59\u5f26\u5b9a\u7406\u3088\u308a\r\n\\[\\begin{align}\r\n\\cos \\angle \\text{AOM} & = \\dfrac{1 +\\frac{3}{4} -\\left( x^2 -\\frac{1}{4} \\right)}{2 \\cdot 1 \\cdot \\frac{\\sqrt{3}}{2}} \\\\\r\n& = \\dfrac{2 -x^2}{\\sqrt{3}}\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\n\\left| \\overrightarrow{a} \\right| & = 1 , \\ \\left| \\overrightarrow{m} \\right| = \\dfrac{\\sqrt{3}}{2} , \\\\\r\n\\overrightarrow{a} \\cdot \\overrightarrow{m} & = 1 \\cdot \\dfrac{\\sqrt{3}}{2} \\cdot \\dfrac{2 -x^2}{\\sqrt{3}} = 1 -\\dfrac{x^2}{2}\r\n\\end{align}\\]\r\n\\(\\overrightarrow{\\text{OH}} = (1-u) \\overrightarrow{a} +u \\overrightarrow{m}\\) \u3068\u304a\u3051\u3070, \\(\\text{OH} \\perp \\text{AM}\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\overrightarrow{\\text{OH}} \\cdot \\overrightarrow{\\text{MA}} & = \\left\\{ (1-u) \\overrightarrow{a} +u \\overrightarrow{m} \\right\\} \\cdot \\left( \\overrightarrow{a} -\\overrightarrow{m} \\right) \\\\\r\n& = 1-u +(2u-1) \\left( 1 -\\dfrac{x^2}{2} \\right) -\\dfrac{3u}{4} \\\\\r\n& = \\dfrac{1}{4} \\left\\{ 4 -4u +4(2 -x^2) u -2( 2 -x^2 ) -3u \\right\\} \\\\\r\n& = \\dfrac{1}{4} \\left\\{ ( 1 -4x^2 ) u +2x^2 \\right\\} = 0 \\\\\r\n& \\text{\u2234} \\quad u = \\dfrac{2x^2}{4x^2 -1}\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\\begin{align}\r\np & = 1-u = \\underline{\\dfrac{2x^2 -1}{4x^2 -1}} , \\\\\r\nq & = r = \\dfrac{u}{2} = \\underline{\\dfrac{x^2}{4x^2 -1}}\r\n\\end{align}\\]\r\n\u307e\u305f, \\(\\overrightarrow{\\text{OH'}} = v \\overrightarrow{m}\\) \u3068\u304a\u3051\u3070, \\(\\text{AH'} \\perp \\text{OM}\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\overrightarrow{\\text{H'A}} \\cdot \\overrightarrow{\\text{OH'}} & = \\left\\{ \\overrightarrow{a} -v \\overrightarrow{m} \\right\\} \\cdot v \\overrightarrow{m} \\\\\r\n& = \\left( 1 -\\dfrac{x^2}{2} \\right) v +\\dfrac{3v^2}{4} \\\\\r\n& = \\dfrac{v}{4} \\left\\{ 3v +2 (2 -x^2) \\right\\} = 0 \\\\\r\n& \\text{\u2234} \\quad v = \\dfrac{2 (x^2 -2)}{3}\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\ns = t = \\dfrac{v}{2} = \\underline{\\dfrac{x^2 -2}{3}}\r\n\\]\r\n

    (2)<\/strong>\r\n\\[\r\n\\sin \\angle \\text{AOM} = \\sqrt{1 -\\cos^2 \\angle \\text{AOM}} = \\sqrt{1 -\\dfrac{(2 -x^2)^2}{3}}\r\n\\]\r\n\u3053\u308c\u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\nV & = \\dfrac{1}{3} \\triangle \\text{OAM} \\cdot \\text{BC} \\\\\r\n& = \\dfrac{1}{3} \\left( \\dfrac{1}{2} \\cdot 1 \\cdot \\dfrac{\\sqrt{3}}{2} \\cdot \\sqrt{1 -\\dfrac{(2 -x^2)^2}{3}} \\right) \\cdot 1 \\\\\r\n& = \\dfrac{\\sqrt{3 -(2 -x^2)^2}}{12} \\quad ... [1] \\\\\r\n& = \\underline{\\dfrac{\\sqrt{-x^4 +4x^2 -1}}{12}}\r\n\\end{align}\\]\r\n\u307e\u305f, [1] \u306e\u5206\u5b50\u306b\u7740\u76ee\u3059\u308c\u3070, \\(V\\) \u304c\u6700\u5927\u3068\u306a\u308b\u306e\u306f, \\(x = \\sqrt{2}\\) \u306e\u3068\u304d\u3067, \u6700\u5927\u5024\u306f\r\n\\[\r\n\\underline{\\dfrac{\\sqrt{3}}{12}}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u56db\u9762\u4f53 OABC \u306b\u304a\u3044\u3066, \\(\\text{OA} = \\text{OB} = \\text{OC} = \\text{BC} = 1\\) , \\(\\text{AB} = \\text{AC} = x\\) \u3068\u3059\u308b. \u9802\u70b9 O … \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":[128],"tags":[141,137],"class_list":["post-1243","post","type-post","status-publish","format-standard","hentry","category-toko_2015","tag-toko","tag-137"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1243","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=1243"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1243\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1243"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1243"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1243"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}