{"id":1311,"date":"2015-10-19T00:47:50","date_gmt":"2015-10-18T15:47:50","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1311"},"modified":"2021-10-20T13:30:48","modified_gmt":"2021-10-20T04:30:48","slug":"ykr201502","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/ykr201502\/","title":{"rendered":"\u6a2a\u56fd\u5927\u7406\u7cfb2015\uff1a\u7b2c2\u554f"},"content":{"rendered":"
\n

\u70b9 O \u3092\u4e2d\u5fc3\u3068\u3059\u308b\u534a\u5f84 \\(1\\) \u306e\u5186\u306b\u5185\u63a5\u3059\u308b\u4e09\u89d2\u5f62 ABC \u304c\u3042\u308a,\r\n\\[\r\n2 \\overrightarrow{\\text{OA}} +3 \\overrightarrow{\\text{OB}} +4 \\overrightarrow{\\text{OC}} = \\overrightarrow{0}\r\n\\]\r\n\u3092\u307f\u305f\u3057\u3066\u3044\u308b. \u3053\u306e\u5186\u4e0a\u306b\u70b9 P \u304c\u3042\u308a, \u7dda\u5206 AB \u3068\u7dda\u5206 CP \u306f\u76f4\u4ea4\u3057\u3066\u3044\u308b. \u6b21\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\u5185\u7a4d \\(\\overrightarrow{\\text{OA}} \\cdot \\overrightarrow{\\text{OB}}\\) \u3068 \\(\\left| \\overrightarrow{\\text{AB}} \\right|\\) \u3092\u305d\u308c\u305e\u308c\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u7dda\u5206 AB \u3068\u7dda\u5206 CP \u306e\u4ea4\u70b9\u3092 H \u3068\u3059\u308b\u3068\u304d, \\(\\text{AH} : \\text{HB}\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\u56db\u89d2\u5f62 APBC \u306e\u9762\u7a4d\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{a} = \\overrightarrow{\\text{OA}}\\) , \\(\\overrightarrow{b} = \\overrightarrow{\\text{OB}}\\) \u3068\u304a\u304f\u3068\r\n\\[\r\n\\overrightarrow{\\text{OC}} = -\\dfrac{1}{2} \\overrightarrow{a} -\\dfrac{3}{4} \\overrightarrow{b}\r\n\\]\r\n\u6761\u4ef6\u3088\u308a, \\(\\left| \\overrightarrow{a} \\right| = \\left| \\overrightarrow{b} \\right| = \\left| \\overrightarrow{\\text{OC}} \\right| = 1\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\left| \\overrightarrow{\\text{OC}} \\right|^2 & = \\dfrac{1}{4} \\left| \\overrightarrow{a} \\right|^2 +\\dfrac{3}{4} \\overrightarrow{a} \\cdot \\overrightarrow{b} +\\dfrac{9}{16} \\left| \\overrightarrow{b} \\right|^2 \\\\\r\n& = \\dfrac{13}{16} +\\dfrac{3}{4} \\overrightarrow{a} \\cdot \\overrightarrow{b} = 1 \\\\\r\n\\text{\u2234} \\quad & \\overrightarrow{a} \\cdot \\overrightarrow{b} = \\underline{\\dfrac{1}{4}}\r\n\\end{align}\\]\r\n\u307e\u305f\r\n\\[\\begin{align}\r\n\\left| \\overrightarrow{\\text{AB}} \\right| & = \\left| \\overrightarrow{b} -\\overrightarrow{a} \\right| \\\\\r\n& = \\sqrt{1 -2 \\cdot \\dfrac{1}{4} +1} \\\\\r\n& = \\underline{\\dfrac{\\sqrt{6}}{2}}\r\n\\end{align}\\]\r\n

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

    \\(\\overrightarrow{\\text{OH}} = t \\overrightarrow{a} +(1-t) \\overrightarrow{b} \\quad ( 0 \\lt t \\lt 1 )\\) \u3068\u304a\u3051\u308b.
    \r\n\\[\\begin{align}\r\n\\overrightarrow{\\text{CH}} & = \\overrightarrow{\\text{OH}} -\\overrightarrow{\\text{OC}} \\\\\r\n& = \\left( t +\\dfrac{1}{2} \\right) \\overrightarrow{a} +\\left( \\dfrac{7}{4} -t \\right) \\overrightarrow{b}\r\n\\end{align}\\]\r\nAB \u306e\u4e2d\u70b9\u3092 M \u3068\u304a\u3051\u3070 \\(\\overrightarrow{\\text{OM}} = \\dfrac{\\overrightarrow{a} +\\overrightarrow{b}}{2}\\) \u3067\u3042\u308a, \\(\\overrightarrow{\\text{CH}} \/ \\hspace{-0.3em} \/ \\overrightarrow{\\text{OM}}\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\nt +\\dfrac{1}{2} & = \\dfrac{7}{4} -t \\\\\r\n\\text{\u2234} \\quad t & = \\dfrac{5}{8}\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n\\overrightarrow{\\text{OH}} = \\dfrac{5}{8} \\overrightarrow{a} +\\dfrac{3}{8} \\overrightarrow{b}\r\n\\]\r\n\u3067\u3042\u308a\r\n\\[\r\n\\text{AH} : \\text{HB} = \\underline{3 : 5}\r\n\\]\r\n

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

    \\[\\begin{align}\r\n\\text{AH} & = \\dfrac{3}{8} \\text{AB} = \\dfrac{3 \\sqrt{6}}{16} , \\\\\r\n\\text{HB} & = \\dfrac{5}{8} \\text{AB} = \\dfrac{5 \\sqrt{6}}{16}\r\n\\end{align}\\]\r\n\u307e\u305f, \\(\\overrightarrow{\\text{CH}} = \\dfrac{9}{8} \\left( \\overrightarrow{a} +\\overrightarrow{b} \\right)\\) \u306a\u306e\u3067\r\n\\[\r\n\\text{CH} = \\dfrac{9}{8} \\sqrt{1 +2 \\cdot \\dfrac{1}{4} +1} = \\dfrac{9 \\sqrt{10}}{16}\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \u65b9\u3079\u304d\u306e\u5b9a\u7406\u3088\u308a\r\n\\[\\begin{align}\r\n\\text{HP} & = \\dfrac{\\text{AH} \\cdot \\text{HB}}{\\text{CH}} \\\\\r\n& = \\dfrac{1}{16} \\cdot \\dfrac{90}{9 \\sqrt{10}} = \\dfrac{\\sqrt{10}}{16}\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u9762\u7a4d \\(S\\) \u306f\r\n\\[\\begin{align}\r\nS & = \\dfrac{1}{2} \\cdot \\text{AB} \\cdot \\text{CP} \\\\\r\n& = \\dfrac{1}{2} \\cdot \\dfrac{\\sqrt{6}}{2} \\cdot \\dfrac{5 \\sqrt{10}}{8} \\\\\r\n& = \\underline{\\dfrac{5 \\sqrt{15}}{16}}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\u70b9 O \u3092\u4e2d\u5fc3\u3068\u3059\u308b\u534a\u5f84 \\(1\\) \u306e\u5186\u306b\u5185\u63a5\u3059\u308b\u4e09\u89d2\u5f62 ABC \u304c\u3042\u308a, \\[ 2 \\overrightarrow{\\text{OA}} +3 \\overrightarrow{\\text{OB}} +4 \\overr … \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":[133],"tags":[137,9],"class_list":["post-1311","post","type-post","status-publish","format-standard","hentry","category-yokokoku_r_2015","tag-137","tag-yokokoku_r"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1311","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=1311"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1311\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1311"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1311"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1311"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}