{"id":4649,"date":"2025-11-09T11:20:13","date_gmt":"2025-11-09T03:20:13","guid":{"rendered":"http:\/\/jhwk.online\/?p=4649"},"modified":"2025-11-09T11:47:50","modified_gmt":"2025-11-09T03:47:50","slug":"%e5%87%bd%e6%95%b0%e5%af%bc%e6%95%b03%e8%ae%b2%e8%a7%a3-3-2-2-2-2","status":"publish","type":"post","link":"http:\/\/jhwk.online\/?p=4649","title":{"rendered":"\u4e09\u89d2\u51fd\u65705"},"content":{"rendered":"\n<div class=\"wp-block-stackable-subtitle stk-block-subtitle stk-block stk-8afe3d6\" data-block-id=\"8afe3d6\"><style>.stk-8afe3d6 .stk-block-subtitle__text{font-size:var(--stk--preset--font-size--medium, 20px) !important;color:#0693e3 !important;}@media screen and (max-width:999px){.stk-8afe3d6 .stk-block-subtitle__text{font-size:var(--stk--preset--font-size--medium, 20px) !important;}}<\/style><p class=\"stk-block-subtitle__text stk-subtitle has-text-color\">\u672c\u9898\u75312023\u7ea722\u73ed\u674e\u4fca\u68a6\u8bb2\u89e3\uff01<\/p><\/div>\n\n\n\n<figure class=\"wp-block-video\"><video controls src=\"http:\/\/jhwk.online\/wp-content\/uploads\/2025\/11\/\u4e09\u89d2\u51fd\u65705.mp4\"><\/video><\/figure>\n\n\n\n<p>[ratemypost]<\/p>\n\n\n\n<hr class=\"wp-block-separator has-text-color has-vivid-cyan-blue-color has-alpha-channel-opacity has-vivid-cyan-blue-background-color has-background is-style-wide\"\/>\n\n\n\n<h5 class=\"wp-block-heading has-vivid-cyan-blue-color has-text-color has-link-color wp-elements-8a052dbd5caa870510f1e594e9e095cf\">\u9898\u76ee\u5c55\u793a\u2014\u2014\u5148\u63a2\u7a76\u3001\u518d\u542c\u8bb2<\/h5>\n\n\n\n\u5728$\\triangle ABC$\u4e2d,\u5185\u89d2$A$,$B$,$C$\u7684\u5bf9\u8fb9\u5206\u522b\u4e3a$a$,$b$,$c$\uff0c\u70b9$O$\u662f$\\triangle ABC$\u7684\u5916\u5fc3,$a\\cos(C &#8211; \\displaystyle \\frac{\\pi}{3}) = \\displaystyle \\frac{\\overrightarrow{AO} \\cdot \\overrightarrow{AB}}{|\\overrightarrow{AB}|} + \\frac{\\overrightarrow{AO} \\cdot \\overrightarrow{AC}}{|\\overrightarrow{AC}|}$.<p>\n\n(1)\u6c42\u89d2$A$;<p>\n\n(2)\u82e5$\\triangle ABC$\u5916\u63a5\u5706\u7684\u5468\u957f\u4e3a$4\\sqrt{3}\\pi$,\u6c42$\\triangle ABC$\u5468\u957f\u7684\u53d6\u503c\u8303\u56f4.\n\n\n\n<h4 class=\"wp-block-heading has-text-align-center has-text-color has-background has-link-color has-medium-font-size wp-elements-58b20738b8080c684d5fff23cf040d12\" style=\"color:#1e4b69;background:linear-gradient(0deg,rgb(230,242,255) 0%,rgb(230,242,255) 16%,rgb(92,175,230) 100%)\">\u89e3\u9898\u63a2\u7a76\u2014\u2014\u70b9\u51fb\u76f4\u8fbe<\/h4>\n\n\n\n<style>\n  .video-jump-btn {\n    cursor: pointer;\n    text-align: left;\n    background-color: #e6f2ff; \/* \u6de1\u84dd\u8272\u80cc\u666f *\/\n    border: none;\n    padding: 8px 12px;\n    border-radius: 4px;\n    transition: all 0.3s ease; \/* \u5e73\u6ed1\u8fc7\u6e21\u52a8\u753b *\/\n    font-size: 16px;\n  }\n  \n  .video-jump-btn:hover {\n    background-color: #cce0ff; \/* \u7565\u6df1\u7684\u84dd\u8272\u80cc\u666f *\/\n    transform: scale(1.02); \/* \u8f7b\u5fae\u653e\u5927\u6548\u679c *\/\n    box-shadow: 0 2px 4px rgba(0,0,0,0.1); \/* \u589e\u52a0\u8f7b\u5fae\u9634\u5f71 *\/\n  }\n<\/style>\n\n<button class=\"video-jump-btn\" onclick=\"document.querySelector('video').currentTime = 0; document.querySelector('video').play();\">\n\u95ee\u9898(1) \u7b2c\u4e00\u6b65\uff1a\u89e3\u91ca$\\displaystyle \\frac{\\overrightarrow{AO} \\cdot \\overrightarrow{AB}}{|\\overrightarrow{AB}|} \u548c\\frac{\\overrightarrow{AO} \\cdot \\overrightarrow{AC}}{|\\overrightarrow{AC}|}$\n\uff1b\n<\/button>\n\n\n\n<br>\n<style>\n  .video-jump-btn {\n    cursor: pointer;\n    text-align: left; \/* \u5de6\u5bf9\u9f50\u6587\u672c *\/\n   \n  }\n<\/style>\n\n<button class=\"video-jump-btn\" onclick=\"document.querySelector('video').currentTime =70; document.querySelector('video').play();\">\n\n \u7b2c\u4e8c\u6b65\uff1a\u6c42\u51fa$A=\\displaystyle\\frac{\\pi}{3}$\uff1b\n\n<\/button>\n\n\n\n<br>\n<style>\n  .video-jump-btn {\n    cursor: pointer;\n    text-align: left; \/* \u5de6\u5bf9\u9f50\u6587\u672c *\/\n   \n  }\n<\/style>\n\n<button class=\"video-jump-btn\" onclick=\"document.querySelector('video').currentTime =148; document.querySelector('video').play();\">\n\n\u95ee\u9898\uff082\uff09 \u65b9\u6cd5\u4e00\u3001\u5229\u7528\u57fa\u672c\u4e0d\u7b49\u5f0f\u6c42\u89e3\uff1b\n\n\n<\/button>\n\n\n\n<br>\n<style>\n  .video-jump-btn {\n    cursor: pointer;\n    text-align: left; \/* \u5de6\u5bf9\u9f50\u6587\u672c *\/\n   \n  }\n<\/style>\n\n<button class=\"video-jump-btn\" onclick=\"document.querySelector('video').currentTime =278; document.querySelector('video').play();\">\n\n \u65b9\u6cd5\u4e8c\u3001\u5229\u7528\u6b63\u5f26\u5b9a\u7406\u6c42\u89e3\u3002\n\n<\/button>\n\n\n\n<details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary>\u7b54\u6848\u89e3\u6790<\/summary>\n \n\n\n\n<p>\u89e3\uff1a(1) \u56e0\u4e3aO\u662f\u25b3ABC\u7684\u5916\u5fc3\uff0c<\/p>\n<p>\u6240\u4ee5\\(\\displaystyle \\frac{\\overrightarrow{AO}\\cdot\\overrightarrow{AB}}{|\\overrightarrow{AB}|}=|\\overrightarrow{AO}|\\cos\\angle OAB = \\displaystyle \\frac{c}{2}\\)\uff0c<\/p>\n<p>\\(\\displaystyle \\frac{\\overrightarrow{AO}\\cdot\\overrightarrow{AC}}{|\\overrightarrow{AC}|}=|\\overrightarrow{AO}|\\cos\\angle OAC = \\displaystyle \\frac{b}{2}\\)\uff0c<\/p>\n<p>\u5219\\(a\\cos(C &#8211; \\displaystyle \\frac{\\pi}{3}) = \\displaystyle \\frac{b + c}{2}\\)\u3002<\/p>\n<p>\u7531\u6b63\u5f26\u5b9a\u7406\uff0c\\(\\sin A\\cos(C &#8211; \\displaystyle \\frac{\\pi}{3}) = \\displaystyle \\frac{\\sin B + \\sin C}{2}\\)\uff0c<\/p>\n<p>\u5c55\u5f00\u5f97\\(\\sin A(\\displaystyle \\frac{1}{2}\\cos C + \\displaystyle \\frac{\\sqrt{3}}{2}\\sin C) = \\displaystyle \\frac{\\sin(A + C) + \\sin C}{2}\\)\uff0c<\/p>\n<p>\u5373\\(\\displaystyle \\frac{1}{2}\\sin A\\cos C + \\displaystyle \\frac{\\sqrt{3}}{2}\\sin A\\sin C = \\displaystyle \\frac{\\sin A\\cos C + \\cos A\\sin C + \\sin C}{2}\\)\uff0c<\/p>\n<p>\u6574\u7406\u5f97\\(\\displaystyle \\frac{\\sqrt{3}}{2}\\sin A\\sin C = \\displaystyle \\frac{1}{2}\\cos A\\sin C + \\displaystyle \\frac{1}{2}\\sin C\\)\uff0c<\/p>\n<p>\u56e0\u4e3a\\(\\sin C \\neq 0\\)\uff0c\u6240\u4ee5\\(\\displaystyle \\frac{\\sqrt{3}}{2}\\sin A = \\displaystyle \\frac{1}{2}\\cos A + \\displaystyle \\frac{1}{2}\\)\uff0c<\/p>\n<p>\u5373\\(\\sin(A &#8211; \\displaystyle \\frac{\\pi}{6}) = \\displaystyle \\frac{1}{2}\\)\u3002<\/p>\n<p>\u56e0\u4e3a\\(A \\in (0, \\pi)\\)\uff0c\u6240\u4ee5\\(A &#8211; \\displaystyle \\frac{\\pi}{6} \\in (-\\displaystyle \\frac{\\pi}{6}, \\displaystyle \\frac{5\\pi}{6})\\)\uff0c<\/p>\n<p>\u6545\\(A &#8211; \\displaystyle \\frac{\\pi}{6} = \\displaystyle \\frac{\\pi}{6}\\)\uff0c\u89e3\u5f97\\(A = \\displaystyle \\frac{\\pi}{3}\\)\u3002<\/p>\n\n<p>(2) \u8bbe\u25b3ABC\u5916\u63a5\u5706\u534a\u5f84\u4e3a\\(R\\)\uff0c\u7531\u5916\u63a5\u5706\u5468\u957f\u4e3a\\(4\\sqrt{3}\\pi\\)\uff0c\u5f97\\(2\\pi R = 4\\sqrt{3}\\pi\\)\uff0c\u89e3\u5f97\\(R = 2\\sqrt{3}\\)\u3002<\/p>\n<p>\u7531\u6b63\u5f26\u5b9a\u7406\uff0c\\(\\displaystyle \\frac{a}{\\sin A} = \\displaystyle \\frac{b}{\\sin B} = \\displaystyle \\frac{c}{\\sin C} = 2R = 4\\sqrt{3}\\)\uff0c<\/p>\n<p>\u6240\u4ee5\\(a = 4\\sqrt{3}\\sin\\displaystyle \\frac{\\pi}{3} = 6\\)\uff0c\\(b = 4\\sqrt{3}\\sin B\\)\uff0c\\(c = 4\\sqrt{3}\\sin C\\)\uff0c\u4e14\\(B + C = \\displaystyle \\frac{2\\pi}{3}\\)\u3002<\/p>\n<p>\u5219\u25b3ABC\u7684\u5468\u957f\\(L = a + b + c = 6 + 4\\sqrt{3}(\\sin B + \\sin C)\\)<\/p>\n<p>\\(= 6 + 4\\sqrt{3}[\\sin B + \\sin(\\displaystyle \\frac{2\\pi}{3} &#8211; B)]\\)<\/p>\n<p>\\(= 6 + 4\\sqrt{3}(\\displaystyle \\frac{3}{2}\\sin B + \\displaystyle \\frac{\\sqrt{3}}{2}\\cos B)\\)<\/p>\n<p>\\(= 6 + 12\\sin(B + \\displaystyle \\frac{\\pi}{6})\\)\u3002<\/p>\n<p>\u56e0\u4e3a\\(B \\in (0, \\displaystyle \\frac{2\\pi}{3})\\)\uff0c\u6240\u4ee5\\(B + \\displaystyle \\frac{\\pi}{6} \\in (\\displaystyle \\frac{\\pi}{6}, \\displaystyle \\frac{5\\pi}{6})\\)\uff0c<\/p>\n<p>\\(\\sin(B + \\displaystyle \\frac{\\pi}{6}) \\in (\\displaystyle \\frac{1}{2}, 1]\\)\uff0c<\/p>\n<p>\u6545\\(L \\in (12, 18]\\)\uff0c\u5373\u25b3ABC\u5468\u957f\u7684\u53d6\u503c\u8303\u56f4\u662f\\((12, 18]\\)\u3002<\/p>\n<\/details>\n","protected":false},"excerpt":{"rendered":"<p>\u672c\u9898\u75312023\u7ea722\u73ed\u674e\u4fca\u68a6\u8bb2\u89e3\uff01 [ratemypost] \u9898\u76ee\u5c55\u793a\u2014\u2014\u5148\u63a2\u7a76\u3001\u518d\u542c\u8bb2 \u5728$\\triangl [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-4649","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"blocksy_meta":[],"_links":{"self":[{"href":"http:\/\/jhwk.online\/index.php?rest_route=\/wp\/v2\/posts\/4649","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/jhwk.online\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/jhwk.online\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/jhwk.online\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/jhwk.online\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=4649"}],"version-history":[{"count":7,"href":"http:\/\/jhwk.online\/index.php?rest_route=\/wp\/v2\/posts\/4649\/revisions"}],"predecessor-version":[{"id":4686,"href":"http:\/\/jhwk.online\/index.php?rest_route=\/wp\/v2\/posts\/4649\/revisions\/4686"}],"wp:attachment":[{"href":"http:\/\/jhwk.online\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4649"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/jhwk.online\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4649"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/jhwk.online\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4649"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}