{"id":2035,"date":"2021-12-15T00:14:32","date_gmt":"2021-12-14T15:14:32","guid":{"rendered":"https:\/\/www.roundown.net\/nyushi\/?p=2035"},"modified":"2021-12-15T00:14:32","modified_gmt":"2021-12-14T15:14:32","slug":"wsr202105","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/wsr202105\/","title":{"rendered":"\u65e9\u7a32\u7530\u7406\u5de52021\uff1a\u7b2c5\u554f"},"content":{"rendered":"
\n

\u6b63\u56db\u9762\u4f53 OABC \u306b\u5bfe\u3057, \u4e09\u89d2\u5f62 ABC \u306e\u5916\u5fc3\u3092 M \u3068\u3057, M \u3092\u4e2d\u5fc3\u3068\u3057\u3066\u70b9 A, B, C \u3092\u901a\u308b\u7403\u9762\u3092 \\(S\\) \u3068\u3059\u308b.\r\n\u307e\u305f, \\(S\\) \u3068\u8fba OA, OB, OC \u3068\u306e\u4ea4\u70b9\u306e\u3046\u3061, A, B, C \u3068\u306f\u7570\u306a\u308b\u3082\u306e\u3092\u305d\u308c\u305e\u308c D, E, F \u3068\u3059\u308b.\r\n\u3055\u3089\u306b, \\(S\\) \u3068\u4e09\u89d2\u5f62 OAB \u306e\u5171\u901a\u90e8\u5206\u3068\u3057\u3066\u5f97\u3089\u308c\u308b\u5f27 DE \u3092\u8003\u3048, \u305d\u306e\u5f27\u3092\u542b\u3080\u5186\u5468\u306e\u4e2d\u5fc3\u3092 G \u3068\u3059\u308b.\r\n\\(\\overrightarrow{a} = \\overrightarrow{\\text{OA}}\\) , \\(\\overrightarrow{b} = \\overrightarrow{\\text{OB}}\\) , \\(\\overrightarrow{c} = \\overrightarrow{\\text{OC}}\\) \u3068\u3057\u3066, \u4ee5\u4e0b\u306e\u554f\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(\\overrightarrow{\\text{OD}} , \\overrightarrow{\\text{OE}} , \\overrightarrow{\\text{OF}} , \\overrightarrow{\\text{OG}}\\) \u3092 \\(\\overrightarrow{a} , \\overrightarrow{b} , \\overrightarrow{c}\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u4e09\u89d2\u5f62 OAB \u306e\u9762\u7a4d\u3092 \\(S_1\\) , \u56db\u89d2\u5f62 ODGE \u306e\u9762\u7a4d\u3092 \\(S_2\\) \u3068\u3059\u308b\u3068\u304d,\r\n\\(S_1 : S_2\\) \u3092\u3067\u304d\u308b\u3060\u3051\u7c21\u5358\u306a\u6574\u6570\u6bd4\u306b\u3088\u308a\u8868\u305b.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

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

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

    \u6b63\u56db\u9762\u4f53\u306e \\(1\\) \u8fba\u306e\u9577\u3055\u3092 \\(1\\) \u3068\u3057\u3066\u3082\u4e00\u822c\u6027\u3092\u5931\u308f\u306a\u3044.
    \r\n\u3053\u306e\u3068\u304d\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 \\dfrac{\\pi}{3} = \\dfrac{1}{2}\r\n\\end{align}\\]\r\n\\(\\triangle \\text{ABC}\\) \u306f\u6b63\u4e09\u89d2\u5f62\u3067\u5916\u5fc3\u3068\u91cd\u5fc3\u304c\u4e00\u81f4\u3059\u308b\u306e\u3067\r\n\\[\r\n\\overrightarrow{\\text{OM}} = \\dfrac{1}{3} \\left( \\overrightarrow{a} +\\overrightarrow{b} +\\overrightarrow{c} \\right)\r\n\\]\r\n\u3086\u3048\u306b\r\n\\[\r\n\\overrightarrow{\\text{AM}} = \\overrightarrow{\\text{OM}} -\\overrightarrow{\\text{OA}} = -\\dfrac{2}{3} \\overrightarrow{a} +\\dfrac{\\overrightarrow{b}}{3} +\\dfrac{\\overrightarrow{c}}{3}\r\n\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\left| \\overrightarrow{\\text{AM}} \\right|^2 & = \\dfrac{4}{9} +2 \\cdot \\dfrac{1}{9} -2 \\cdot \\dfrac{4}{9} \\cdot \\dfrac{1}{2} +\\dfrac{2}{9} \\cdot \\dfrac{1}{2} \\\\\r\n& = \\dfrac{6 -4 +1}{9} = \\dfrac{1}{3}\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(S\\) \u306e\u534a\u5f84 \\(R\\) \u306f, \\(R = \\dfrac{1}{\\sqrt{3}}\\) .
    \r\n\\(\\overrightarrow{\\text{OD}} = t \\overrightarrow{a} \\ ( 0 \\lt t \\lt 1 )\\) \u3068\u304a\u3051\u3070\r\n\\[\r\n\\overrightarrow{\\text{DM}} = \\overrightarrow{\\text{OM}} -\\overrightarrow{\\text{OD}} = \\left( \\dfrac{1}{3} -t \\right) \\overrightarrow{a} +\\dfrac{\\overrightarrow{b}}{3} +\\dfrac{\\overrightarrow{c}}{3}\r\n\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\left| \\overrightarrow{\\text{DM}} \\right|^2 =\r\n\\left( \\dfrac{1}{3} -t \\right)^2 + & 2 \\cdot \\dfrac{1}{9} -2 \\cdot \\dfrac{1 -3t}{9} +\\dfrac{1}{9} \\\\\r\n= \\dfrac{2}{3} -\\dfrac{4}{3} t +t^2 & = \\dfrac{1}{3} \\\\\r\n\\text{\u2234} \\quad 3t^2 -4t +1 & = 0 \\\\\r\n( 3t -1 ) (t-1) & = 0 \\\\\r\n\\text{\u2234} \\quad t & = \\dfrac{1}{3}\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n\\overrightarrow{\\text{OD}} = \\underline{\\dfrac{1}{3} \\overrightarrow{a}}\r\n\\]\r\nE, F \u306b\u3064\u3044\u3066\u3082\u540c\u69d8\u306b\u8003\u3048\u3066\r\n\\[\r\n\\overrightarrow{\\text{OE}} = \\underline{\\dfrac{1}{3} \\overrightarrow{b}} \\ , \\ \\overrightarrow{\\text{OF}} = \\underline{\\dfrac{1}{3} \\overrightarrow{c}}\r\n\\]\r\nA, B, D, E \u306f\u540c\u4e00\u5186\u5468\u4e0a\u306b\u3042\u308a, \u5bfe\u79f0\u6027\u304b\u3089 \\(\\overrightarrow{\\text{OG}} = s \\left( \\overrightarrow{a} +\\overrightarrow{b} \\right)\\) \u3068\u304a\u3051\u308b.\r\n\\[\\begin{align}\r\n\\overrightarrow{\\text{AG}} & = \\overrightarrow{\\text{OG}} -\\overrightarrow{\\text{OA}} = (s-1) \\overrightarrow{a} +s \\overrightarrow{b} \\ , \\\\\r\n\\left| \\overrightarrow{\\text{AG}} \\right|^2 & = (s-1)^2 +s^2 +s (s-1) \\\\\r\n& = 3 s^2 -3s +1\r\n\\end{align}\\]\r\n\u307e\u305f\r\n\\[\\begin{align}\r\n\\overrightarrow{\\text{DG}} & = \\overrightarrow{\\text{OG}} -\\overrightarrow{\\text{OD}} = \\left( s -\\dfrac{1}{3} \\right) \\overrightarrow{a} +s \\overrightarrow{b} \\ , \\\\\r\n\\left| \\overrightarrow{\\text{DG}} \\right|^2 & = \\left( s -\\dfrac{1}{3} \\right)^2 +s^2 +s \\left( s -\\dfrac{1}{3} \\right) \\\\\r\n& = 3 s^2 -s +\\dfrac{1}{9}\r\n\\end{align}\\]\r\n\\(\\left| \\overrightarrow{\\text{AG}} \\right| = \\left| \\overrightarrow{\\text{DG}} \\right|\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\n3 s^2 -3s +1 & = 3s^2 -s +\\dfrac{1}{9} \\\\\r\n2s & = \\dfrac{8}{9} \\\\\r\n\\text{\u2234} \\quad s & = \\dfrac{4}{9}\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n\\overrightarrow{\\text{OG}} = \\underline{\\dfrac{4}{9} \\left( \\overrightarrow{a} +\\overrightarrow{b} \\right)}\r\n\\]\r\n

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

    AB \u306e\u4e2d\u70b9\u3092 N \u3068\u304a\u304f.
    \r\n\u5bfe\u79f0\u6027\u304b\u3089, \\(\\triangle \\text{OAN} : \\triangle \\text{ODG}\\) \u3092\u8003\u3048\u308c\u3070\u3088\u3044.\r\n\\[\r\n\\text{OA} : \\text{OD} = 3 : 1\r\n\\]\r\n\\(\\overrightarrow{\\text{OG}} = \\dfrac{8}{9} \\overrightarrow{\\text{ON}}\\) \u306a\u306e\u3067\r\n\\[\r\n\\text{ON} : \\text{OG} = 9 : 8\r\n\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u6bd4\u306f\r\n\\[\r\nS_1 : S_2 = 3 \\cdot 9 : 1 \\cdot 8 = \\underline{27 : 8}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u6b63\u56db\u9762\u4f53 OABC \u306b\u5bfe\u3057, \u4e09\u89d2\u5f62 ABC \u306e\u5916\u5fc3\u3092 M \u3068\u3057, M \u3092\u4e2d\u5fc3\u3068\u3057\u3066\u70b9 A, B, C \u3092\u901a\u308b\u7403\u9762\u3092 \\(S\\) \u3068\u3059\u308b. \u307e\u305f, \\(S\\) \u3068\u8fba OA, OB, OC \u3068\u306e\u4ea4\u70b9\u306e\u3046\u3061, A, B, C … \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":[177],"tags":[147,165],"class_list":["post-2035","post","type-post","status-publish","format-standard","hentry","category-waseda_r_2021","tag-waseda_r","tag-165"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/2035","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=2035"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/2035\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=2035"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=2035"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=2035"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}