{"id":1131,"date":"2023-08-14T22:25:01","date_gmt":"2023-08-14T19:25:01","guid":{"rendered":"https:\/\/bagrut-ru.com\/?p=1131"},"modified":"2025-09-04T17:43:41","modified_gmt":"2025-09-04T14:43:41","slug":"%d0%b1%d0%b0%d0%b3%d1%80%d1%83%d1%82-5-%d0%b7%d0%b8%d0%bc%d0%b0-2023-%d0%b2%d0%b0%d1%80%d0%b8%d0%b0%d0%bd%d1%82-35571-%d0%b7%d0%b0%d0%b4%d0%b0%d0%bd%d0%b8%d0%b5-5","status":"publish","type":"post","link":"https:\/\/bagrut-ru.com\/?p=1131","title":{"rendered":"\u0411\u0410\u0413\u0420\u0423\u0422 (5). \u0417\u0418\u041c\u0410 2023, \u0412\u0410\u0420\u0418\u0410\u041d\u0422 35571, \u0417\u0410\u0414\u0410\u041d\u0418\u0415 5"},"content":{"rendered":"\n
\n
\"\"<\/figure>\n<\/figure>\n\n\n\n
\n
\"\"<\/figure>\n<\/figure>\n\n\n\n

\u0438\u0441\u0442\u043e\u0447\u043d\u0438\u043a<\/a><\/p>\n\n\n\n

\n\t\t\t\t
\u0440\u0435\u0448\u0435\u043d\u0438\u0435<\/div>\n\t\t\t\t
<\/p>\n\n\n\n
\n
\n

(\u05d0)<\/mark><\/strong> \u0420\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0438\u043c \u0442\u0440\u0435\u0443\u0433\u043e\u043b\u044c\u043d\u0438\u043a \\(BCD\\). \\(\\measuredangle BCD=90^{\\circ}\\), \u0442\u0430\u043a \u043a\u0430\u043a \u0432\u043f\u0438\u0441\u0430\u043d\u043d\u044b\u0439 \u0443\u0433\u043e\u043b, \u043e\u043f\u0438\u0440\u0430\u044e\u0449\u0438\u0439\u0441\u044f \u043d\u0430 \u0434\u0438\u0430\u043c\u0435\u0442\u0440 \u0440\u0430\u0432\u0435\u043d \\(90^{\\circ}\\). (\\(BD\\) — \u0434\u0438\u0430\u043c\u0435\u0442\u0440, \\(BD=2R\\)).<\/p>\n\n\n\n

\u041a\u043e\u0441\u0438\u043d\u0443\u0441 \u043e\u0441\u0442\u0440\u043e\u0433\u043e \u0443\u0433\u043b\u0430 \u043f\u0440\u044f\u043c\u043e\u0443\u0433\u043e\u043b\u044c\u043d\u043e\u0433\u043e \u0442\u0440\u0435\u0443\u0433\u043e\u043b\u044c\u043d\u0438\u043a\u0430 \u0440\u0430\u0432\u0435\u043d \u043e\u0442\u043d\u043e\u0448\u0435\u043d\u0438\u044e \u043f\u0440\u0438\u043b\u0435\u0436\u0430\u0449\u0435\u0433\u043e \u043a\u0430\u0442\u0435\u0442\u0430 \u043a \u0433\u0438\u043f\u043e\u0442\u0435\u043d\u0443\u0437\u0435. \u041f\u043e\u044d\u0442\u043e\u043c\u0443 \\(\\cos \\alpha=\\frac{CD}{BD}=\\frac{m}{2R}\\)<\/p>\n\n\n\n

\u041e\u0442\u0432\u0435\u0442: <\/em>\\(\\cos \\alpha=\\frac{m}{2R}\\)<\/p>\n\n\n\n

(\u05d1)<\/mark><\/strong> \u0420\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0438\u043c \u0442\u0440\u0435\u0443\u0433\u043e\u043b\u044c\u043d\u0438\u043a \\(CED\\). \u041f\u043e \u0443\u0441\u043b\u043e\u0432\u0438\u044e \\(OE=ED\\), \u043f\u043e\u044d\u0442\u043e\u043c\u0443 \\(ED=\\frac{OD}{2}=\\frac{R}{2}\\). <\/p>\n\n\n\n

\u0427\u0442\u043e\u0431\u044b \u043d\u0430\u0439\u0442\u0438 \\(CE\\) \u0432\u043e\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u0435\u043c\u0441\u044f \u0442\u0435\u043e\u0440\u0435\u043c\u043e\u0439 \u043a\u043e\u0441\u0438\u043d\u0443\u0441\u043e\u0432: \u043a\u0432\u0430\u0434\u0440\u0430\u0442 \u0441\u0442\u043e\u0440\u043e\u043d\u044b \u0442\u0440\u0435\u0443\u0433\u043e\u043b\u044c\u043d\u0438\u043a\u0430 \u0440\u0430\u0432\u0435\u043d \u0441\u0443\u043c\u043c\u0435 \u043a\u0432\u0430\u0434\u0440\u0430\u0442\u043e\u0432 \u0434\u0432\u0443\u0445 \u0434\u0440\u0443\u0433\u0438\u0445 \u0441\u0442\u043e\u0440\u043e\u043d \u043c\u0438\u043d\u0443\u0441 \u0443\u0434\u0432\u043e\u0435\u043d\u043d\u043e\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u0435 \u044d\u0442\u0438\u0445 \u0441\u0442\u043e\u0440\u043e\u043d \u043d\u0430 \u043a\u043e\u0441\u0438\u043d\u0443\u0441 \u0443\u0433\u043b\u0430 \u043c\u0435\u0436\u0434\u0443 \u043d\u0438\u043c\u0438. <\/p>\n\n\n\n

\\(CE^2=CD^2+ED^2-2\\cdot CD\\cdot ED\\cdot \\cos \\alpha\\)<\/p>\n\n\n\n

\\(CE^2=m^2+(\\frac{R}{2})^2-2\\cdot m\\cdot \\frac{R}{2}\\cdot \\cos \\alpha\\)<\/p>\n\n\n\n

\u0418\u0437 \u043f. (\u05d0)<\/mark><\/strong> <\/em>\\(\\cos \\alpha=\\frac{m}{2R}\\), \u0442\u043e\u0433\u0434\u0430 <\/p>\n\n\n\n

\\(CE^2=m^2+(\\frac{R}{2})^2-2\\cdot m\\cdot \\frac{R}{2}\\cdot \\frac{m}{2R}=\\)<\/p>\n\n\n\n

\\(=m^2+\\frac{R^2}{4}-\\frac{m^2}{2}=\\frac{2m^2+R^2}{4}\\)<\/p>\n\n\n\n

\\(CE=\\sqrt{\\frac{2m^2+R^2}{4}}=\\frac{1}{2}\\sqrt{2m^2+R^2}\\)<\/p>\n\n\n\n

\u0427\u0442\u043e \u0438 \u0442\u0440\u0435\u0431\u043e\u0432\u0430\u043b\u043e\u0441\u044c \u0434\u043e\u043a\u0430\u0437\u0430\u0442\u044c.<\/p>\n\n\n\n

(\u05d2)<\/mark><\/strong> \u0412\u044b\u0447\u0438\u0441\u043b\u0438\u043c \\(\\alpha\\), \u0435\u0441\u043b\u0438 \\(BC=EC\\).<\/p>\n\n\n\n

\u0420\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0438\u043c \u0442\u0440\u0435\u0443\u0433\u043e\u043b\u044c\u043d\u0438\u043a \\(BCE\\). \u0422\u0430\u043a \u043a\u0430\u043a \u043f\u043e \u0443\u0441\u043b\u043e\u0432\u0438\u044e \\(BC=EC\\), \u044d\u0442\u043e\u0442 \u0442\u0440\u0435\u0443\u0433\u043e\u043b\u044c\u043d\u0438\u043a \u0440\u0430\u0432\u043d\u043e\u0431\u0435\u0434\u0440\u0435\u043d\u043d\u044b\u0439. \u041f\u0440\u043e\u0432\u0435\u0434\u0435\u043c \u0432\u044b\u0441\u043e\u0442\u0443 \\(CK\\). \u0412 \u0440\u0430\u0432\u043d\u043e\u0431\u0435\u0434\u0440\u0435\u043d\u043d\u043e\u043c \u0442\u0440\u0435\u0443\u0433\u043e\u043b\u044c\u043d\u0438\u043a\u0430 \u0432\u044b\u0441\u043e\u0442 \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u043c\u0435\u0434\u0438\u0430\u043d\u043e\u0439, \u043f\u043e\u044d\u0442\u043e\u043c\u0443 \\(BK=KE=\\frac{BE}{2}=\\frac{BD-ED}{2}=\\frac{2R-\\frac{R}{2}}{2}=\\frac{3R}{4}\\). <\/p>\n\n\n\n

\u0420\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0438\u043c \u043f\u0440\u044f\u043c\u043e\u0443\u0433\u043e\u043b\u044c\u043d\u044b\u0439 \u0442\u0440\u0435\u0443\u0433\u043e\u043b\u044c\u043d\u0438\u043a \\(CBK\\).<\/p>\n\n\n\n

\\(\\measuredangle B=90^{\\circ}-\\measuredangle D=90^{\\circ}-\\alpha\\)- \u0438\u0437 \u043f\u0440\u044f\u043c\u043e\u0443\u0433\u043e\u043b\u044c\u043d\u043e\u0433\u043e \u0442\u0440\u0435\u0443\u0433\u043e\u043b\u044c\u043d\u0438\u043a\u0430 \\(BCD\\), \u0442\u0430\u043a \u043a\u0430\u043a \u0441\u0443\u043c\u043c\u0430 \u043e\u0441\u0442\u0440\u044b\u0445 \u0443\u0433\u043b\u043e\u0432 \u0432 \u043f\u0440\u044f\u043c\u043e\u0443\u0433\u043e\u043b\u044c\u043d\u043e\u043c \u0442\u0440\u0435\u0443\u0433\u043e\u043b\u044c\u043d\u0438\u043a\u0435 \u0440\u0430\u0432\u043d\u0430 \\(90^{\\circ}\\)<\/p>\n<\/div>\n\n\n\n

\n
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\"\"<\/figure>\n<\/figure>\n\n\n\n

\\(\\\\\\)<\/p>\n\n\n\n

\n
\"\"<\/figure>\n<\/figure>\n\n\n\n

\\(\\\\\\)<\/p>\n\n\n\n

\\(\\\\\\)<\/p>\n\n\n\n

\\(\\\\\\)<\/p>\n\n\n\n

\\(\\\\\\)<\/p>\n\n\n\n

\n
\"\"<\/figure>\n<\/figure>\n\n\n\n

\\(\\\\\\)<\/p>\n\n\n\n

<\/p>\n\n\n\n

<\/p>\n<\/div>\n<\/div>\n\n\n\n

\u0422\u043e\u0433\u0434\u0430 \\(\\cos \\measuredangle B=\\frac{BK}{BC}\\)<\/p>\n\n\n\n

\u0422\u0430\u043a \u043a\u0430\u043a \\(BC=EC\\) \u043f\u043e \u0443\u0441\u043b\u043e\u0432\u0438\u044e, \\(CE=\\frac{1}{2}\\sqrt{2m^2+R^2}\\) (\u0438\u0437 \u043f. (\u05d1)<\/mark><\/strong>), \\(BK=\\frac{3R}{4}\\), \\(\\cos (90^{\\circ}-\\alpha)=\\sin \\alpha\\), \u043f\u043e\u043b\u0443\u0447\u0438\u043c<\/p>\n\n\n\n

\\(\\sin \\alpha=\\frac {\\frac{3R}{4}}{\\frac{1}{2}\\sqrt{2m^2+R^2}}\\).<\/p>\n\n\n\n

\u041d\u0430\u0439\u0434\u0435\u043c, \u043a\u0430\u043a \u0441\u043e\u043e\u0442\u043d\u043e\u0441\u044f\u0442\u0441\u044f \\(m\\) \u0438 \\(R\\).<\/p>\n\n\n\n

\u0417\u0430\u043f\u0438\u0448\u0435\u043c \u0442\u0435\u043e\u0440\u0435\u043c\u0443 \u041f\u0438\u0444\u0430\u0433\u043e\u0440\u0430 \u0434\u043b\u044f \u0442\u0440\u0435\u0443\u0433\u043e\u043b\u044c\u043d\u0438\u043a\u0430 \\(BCD\\):<\/p>\n\n\n\n

\\(BD^2=BC^2+CD^2\\)<\/p>\n\n\n\n

\\((2R)^2=(\\frac{1}{2}\\sqrt{2m^2+R^2})^2+m^2\\)<\/p>\n\n\n\n

\\(4R^2=\\frac{2m^2+R^2}{4}+m^2\\)<\/p>\n\n\n\n

\\(16R^2=6m^2+R^2\\), \\(15R^2=6m^2\\), \\(2m^2=5R^2\\)<\/mark><\/p>\n\n\n\n

\u0422\u043e\u0433\u0434\u0430 \\(\\sin \\alpha=\\frac {\\frac{3R}{4}}{\\frac{1}{2}\\sqrt{2m^2+R^2}}=\\frac {\\frac{3R}{4}}{\\frac{1}{2}\\sqrt{5R^2+R^2}}=\\frac{\\frac{3R}{2}}{R\\sqrt 6}=\\frac{3}{2\\sqrt 6}\\)<\/p>\n\n\n\n

\\(\\sin \\alpha=\\frac{3}{2\\sqrt 6}\\)<\/mark><\/p>\n\n\n\n

\\(\\alpha=\\arcsin \\frac{3}{2\\sqrt 6}\\approx 37.76^{\\circ}\\)<\/p>\n\n\n\n

\u041e\u0442\u0432\u0435\u0442: \\(\\approx 37.76^{\\circ}\\)<\/p>\n\n\n\n

\n
\n

(\u05d3)<\/mark><\/strong> \u0420\u0430\u0434\u0438\u0443\u0441 \\(OG\\) \u043f\u0430\u0440\u0430\u043b\u043b\u0435\u043b\u0435\u043d \\(CD\\) \u043f\u043e \u0443\u0441\u043b\u043e\u0432\u0438\u044e, \u0441\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e, \\(\\measuredangle EOG=\\measuredangle ODC=\\alpha\\) \u043a\u0430\u043a \u043d\u0430\u043a\u0440\u0435\u0441\u0442 \u043b\u0435\u0436\u0430\u0449\u0438\u0435. \\(OE=ED=\\frac{R}{2}\\), \\(OG=R\\).<\/p>\n\n\n\n

\u0420\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0438\u043c \u0442\u0440\u0435\u0443\u0433\u043e\u043b\u044c\u043d\u0438\u043a \\(OEG\\). \u0421\u043d\u0430\u0447\u0430\u043b\u0430 \u043f\u043e \u0442\u0435\u043e\u0440\u0435\u043c\u0435 \u043a\u043e\u0441\u0438\u043d\u0443\u0441\u043e\u0432 \u043d\u0430\u0439\u0434\u0435\u043c \\(EG\\), \u0430 \u0437\u0430\u0442\u0435\u043c \u043f\u043e \u0442\u0435\u043e\u0440\u0435\u043c\u0435 \u0441\u0438\u043d\u0443\u0441\u043e\u0432 \u043d\u0430\u0439\u0434\u0435\u043c \u0441\u0438\u043d\u0443\u0441 \u0443\u0433\u043b\u0430 \\(OEG\\).<\/p>\n\n\n\n

\\(EG^2=OE^2+OG^2-2\\cdot OE \\cdot OG\\cdot \\cos \\alpha\\)<\/p>\n\n\n\n

\\(\\cos ^2 \\alpha=1-\\sin^2 \\alpha=1-(\\frac{3}{2\\sqrt{6}})^2=1-\\frac{9}{24}=\\)<\/p>\n\n\n\n

\\(=\\frac{15}{24}=\\frac{5}{8}\\)<\/p>\n\n\n\n

\\(\\cos \\alpha=\\sqrt{\\frac{5}{8}}\\)<\/p>\n<\/div>\n\n\n\n

\n
\n
\"\"<\/figure>\n<\/figure>\n<\/div>\n<\/div>\n\n\n\n

\u0422\u043e\u0433\u0434\u0430: \\(EG^2=(\\frac{R}{2})^2+R^2-2\\cdot \\frac{R}{2} \\cdot R\\cdot \\sqrt{\\frac{5}{8}}=R^2(\\frac{5}{4}-\\sqrt{\\frac{5}{8}})\\)<\/p>\n\n\n\n

\\(EG=R\\sqrt {\\frac{5}{4}-\\sqrt{\\frac{5}{8}}}\\)<\/p>\n\n\n\n

\u0422\u0435\u043f\u0435\u0440\u044c \u043c\u043e\u0436\u0435\u043c \u0437\u0430\u043f\u0438\u0441\u0430\u0442\u044c \u0442\u0435\u043e\u0440\u0435\u043c\u0443 \u0441\u0438\u043d\u0443\u0441\u043e\u0432 \u0434\u043b\u044f \u0442\u0440\u0435\u0443\u0433\u043e\u043b\u044c\u043d\u0438\u043a\u0430 \\(OEG\\)<\/p>\n\n\n\n

\\(\\frac{OG}{\\sin OEG}=\\frac{EG}{\\sin \\alpha}\\)<\/p>\n\n\n\n

\\(\\frac{R}{\\sin OEG}=\\frac{R\\sqrt {\\frac{5}{4}-\\sqrt{\\frac{5}{8}}}}{\\frac{3}{2\\sqrt{6}}}\\)<\/p>\n\n\n\n

\\(\\sin OEG=\\frac{R \\cdot \\frac{3}{2\\sqrt{6}}}{R\\sqrt {\\frac{5}{4}-\\sqrt{\\frac{5}{8}}}}=\\frac{3}{2\\sqrt{6}\\sqrt {\\frac{5}{4}-\\sqrt{\\frac{5}{8}}}}\\approx 0.903453\\)<\/p>\n\n\n\n

\\(\\arcsin 0.903453\\approx 64.62^{\\circ}\\)<\/p>\n\n\n\n

\u0422\u0430\u043a \u043a\u0430\u043a \u0443\u0433\u043e\u043b \\(OEG\\) — \u0442\u0443\u043f\u043e\u0439, \\(\\measuredangle OEG=180^{\\circ}-64.62^{\\circ}=115.38^{\\circ}\\)<\/p>\n\n\n\n

\u041e\u0442\u0432\u0435\u0442: \\(\\measuredangle OEG=115.38^{\\circ}\\)<\/p>\n\n\n\n

<\/div>\n\t\t\t<\/div><\/p>\n","protected":false},"excerpt":{"rendered":"

\u0438\u0441\u0442\u043e\u0447\u043d\u0438\u043a<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[7],"tags":[],"class_list":["post-1131","post","type-post","status-publish","format-standard","hentry","category-7"],"_links":{"self":[{"href":"https:\/\/bagrut-ru.com\/index.php?rest_route=\/wp\/v2\/posts\/1131","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/bagrut-ru.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/bagrut-ru.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/bagrut-ru.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/bagrut-ru.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1131"}],"version-history":[{"count":59,"href":"https:\/\/bagrut-ru.com\/index.php?rest_route=\/wp\/v2\/posts\/1131\/revisions"}],"predecessor-version":[{"id":1320,"href":"https:\/\/bagrut-ru.com\/index.php?rest_route=\/wp\/v2\/posts\/1131\/revisions\/1320"}],"wp:attachment":[{"href":"https:\/\/bagrut-ru.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1131"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/bagrut-ru.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1131"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/bagrut-ru.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1131"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}