{"id":387,"date":"2023-06-29T17:45:26","date_gmt":"2023-06-29T14:45:26","guid":{"rendered":"https:\/\/bagrut-ru.com\/?p=387"},"modified":"2025-09-04T17:44:34","modified_gmt":"2025-09-04T14:44:34","slug":"%d0%b1%d0%b0%d0%b3%d1%80%d1%83%d1%82-5-%d0%bb%d0%b5%d1%82%d0%be-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=387","title":{"rendered":"\u0411\u0410\u0413\u0420\u0423\u0422 (5). \u041b\u0415\u0422\u041e 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\t\t\t\t
\u0440\u0435\u0448\u0435\u043d\u0438\u0435<\/div>\n\t\t\t\t
<\/p>\n\n\n\n

\u0412\u0441\u043f\u043e\u043c\u043d\u0438\u043c: \u0434\u0435\u043b\u044c\u0442\u043e\u0438\u0434 — \u044d\u0442\u043e \u0447\u0435\u0442\u044b\u0440\u0435\u0445\u0443\u0433\u043e\u043b\u044c\u043d\u0438\u043a, \u0443 \u043a\u043e\u0442\u043e\u0440\u043e\u0433\u043e \u0434\u0432\u0435 \u043f\u0430\u0440\u044b \u0440\u0430\u0432\u043d\u044b\u0445 \u0441\u043c\u0435\u0436\u043d\u044b\u0445 \u0441\u0442\u043e\u0440\u043e\u043d. <\/p>\n\n\n\n

1. \u0414\u0438\u0430\u0433\u043e\u043d\u0430\u043b\u0438 \u0434\u0435\u043b\u044c\u0442\u043e\u0438\u0434\u0430 \u043f\u0435\u0440\u043f\u0435\u043d\u0434\u0438\u043a\u0443\u043b\u044f\u0440\u043d\u044b.<\/p>\n\n\n\n

2. \u0413\u043b\u0430\u0432\u043d\u0430\u044f \u0434\u0438\u0430\u0433\u043e\u043d\u0430\u043b\u044c \u0434\u0435\u043b\u044c\u0442\u043e\u0438\u0434\u0430 \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u0431\u0438\u0441\u0441\u0435\u043a\u0442\u0440\u0438\u0441\u043e\u0439 \u0443\u0433\u043b\u043e\u0432. <\/p>\n\n\n\n

3. \u0422\u043e\u0447\u043a\u0430 \u043f\u0435\u0440\u0435\u0441\u0435\u0447\u0435\u043d\u0438\u044f \u0434\u0438\u0430\u0433\u043e\u043d\u0430\u043b\u0435\u0439 \u0434\u0435\u043b\u0438\u0442 \u0432\u0442\u043e\u0440\u0443\u044e \u0434\u0438\u0430\u0433\u043e\u043d\u0430\u043b\u044c \u043f\u043e\u043f\u043e\u043b\u0430\u043c. <\/p>\n\n\n\n

4. \u0413\u043b\u0430\u0432\u043d\u0430\u044f \u0434\u0438\u0430\u0433\u043e\u043d\u0430\u043b\u044c \u0434\u0435\u043b\u044c\u0442\u043e\u0438\u0434\u0430 \u044f\u0432\u043b\u044f\u0435\u0442\u0441\u044f \u0435\u0433\u043e \u043e\u0441\u044c\u044e \u0441\u0438\u043c\u043c\u0435\u0442\u0440\u0438\u0438. <\/p>\n\n\n\n

\n
\n

(\u05d0)<\/mark><\/strong> (1) <\/mark>\u041f\u043e \u0443\u0441\u043b\u043e\u0432\u0438\u044e \u0437\u0430\u0434\u0430\u0447\u0438 \u0434\u0435\u043b\u044c\u0442\u043e\u0438\u0434 \u0432\u043f\u0438\u0441\u0430\u043d \u0432 \u043e\u043a\u0440\u0443\u0436\u043d\u043e\u0441\u0442\u044c, \u0441\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e, \u0438\u0437 \u0441\u043e\u043e\u0431\u0440\u0430\u0436\u0435\u043d\u0438\u0439 \u0441\u0438\u043c\u043c\u0435\u0442\u0440\u0438\u0438, \u0446\u0435\u043d\u0442\u0440 \u043e\u043f\u0438\u0441\u0430\u043d\u043d\u043e\u0439 \u043e\u043a\u0440\u0443\u0436\u043d\u043e\u0441\u0442\u0438 \u043b\u0435\u0436\u0438\u0442 \u0432 \u0441\u0435\u0440\u0435\u0434\u0438\u043d\u0435 \u0433\u043b\u0430\u0432\u043d\u043e\u0439 \u0434\u0438\u0430\u0433\u043e\u043d\u0430\u043b\u0438. \u0422\u043e\u0433\u0434\u0430 \u0443\u0433\u043e\u043b \u0434\u0435\u043b\u044c\u0442\u043e\u0438\u0434\u0430, \u043a\u043e\u0442\u043e\u0440\u044b\u0439 \u043e\u043f\u0438\u0440\u0430\u0435\u0442\u0441\u044f \u043d\u0430 \u0433\u043b\u0430\u0432\u043d\u0443\u044e \u0434\u0438\u0430\u0433\u043e\u043d\u0430\u043b\u044c \u0440\u0430\u0432\u0435\u043d \\(90^{\\circ}\\), \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. \\(\\angle ABC=90^{\\circ}\\)<\/p>\n\n\n\n

\u041f\u043e \u0443\u0441\u043b\u043e\u0432\u0438\u044e \\( \\measuredangle CAB=\\alpha \\). \u0422\u043e\u0433\u0434\u0430 \\( \\measuredangle ACB =90^{\\circ}-\\alpha \\), \u0442\u0430\u043a \u043a\u0430\u043a \u0441\u0443\u043c\u043c\u0430 \u043e\u0441\u0442\u0440\u044b\u0445 \u0443\u0433\u043b\u043e\u0432 \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 \\(ABC\\) \u0440\u0430\u0432\u043d\u0430 \\(\\ 180^{\\circ}\\)<\/p>\n\n\n\n

\u0426\u0435\u043d\u0442\u0440 \u043e\u043a\u0440\u0443\u0436\u043d\u043e\u0441\u0442\u0438, \u0432\u043f\u0438\u0441\u0430\u043d\u043d\u043e\u0439 \u0432 \u0442\u0440\u0435\u0443\u0433\u043e\u043b\u044c\u043d\u0438\u043a \u043b\u0435\u0436\u0438\u0442 \u0432 \u0442\u043e\u0447\u043a\u0435 \u043f\u0435\u0440\u0435\u0441\u0435\u0447\u0435\u043d\u0438\u044f \u0431\u0438\u0441\u0441\u0435\u043a\u0442\u0440\u0438\u0441 \u0442\u0440\u0435\u0443\u0433\u043e\u043b\u044c\u043d\u0438\u043a\u0430. \u0422\u043e \u0435\u0441\u0442\u044c \\(AO\\) \u0438 \\(CO\\) — \u0431\u0438\u0441\u0441\u0435\u043a\u0442\u0440\u0438\u0441\u044b \u0442\u0440\u0435\u0443\u0433\u043e\u043b\u044c\u043d\u0438\u043a\u0430 \\(ABC\\).<\/p>\n\n\n\n

\u041e\u0442\u0441\u044e\u0434\u0430 \\( \\measuredangle CAO=\\frac{\\alpha}{2}\\), \\(\\measuredangle OCA=\\frac{90^{\\circ}-\\alpha}{2}=45^{\\circ}-\\frac{\\alpha}{2}\\), \\(\\measuredangle AOC =180^{\\circ}-\\frac{\\alpha}{2}-(45^{\\circ}-\\frac{\\alpha}{2})=135^{\\circ}\\).<\/p>\n<\/div>\n\n\n\n

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

(\u05d0)<\/strong> (2)<\/mark> \u0412\u044b\u0440\u0430\u0437\u0438\u043c \u0434\u043b\u0438\u043d\u0443 \u043e\u0442\u0440\u0435\u0437\u043a\u0430 \\(AO\\). \u0414\u043b\u044f \u044d\u0442\u043e\u0433\u043e \u0440\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0438\u043c \u0442\u0440\u0435\u0443\u0433\u043e\u043b\u044c\u043d\u0438\u043a \\(AOC\\). \u0412 \u043d\u0435\u043c \u043d\u0430\u043c \u0438\u0437\u0432\u0435\u0441\u0442\u043d\u0430 \u0441\u0442\u043e\u0440\u043e\u043d\u0430 \\(AC=2R\\) \u0438 \u0442\u0440\u0438 \u0443\u0433\u043b\u0430. \u0412 \u044d\u0442\u043e\u043c \u0441\u043b\u0443\u0447\u0430\u0435, \u0447\u0442\u043e\u0431\u044b \u043d\u0430\u0439\u0442\u0438 \u043b\u044e\u0431\u0443\u044e \u0434\u0440\u0443\u0433\u0443\u044e \u0441\u0442\u043e\u0440\u043e\u043d\u0443 \u0442\u0440\u0435\u0443\u0433\u043e\u043b\u044c\u043d\u0438\u043a\u0430, \u043c\u044b \u0432\u043e\u0441\u043f\u043e\u043b\u044c\u0437\u0443\u0435\u043c\u0441\u044f \u0442\u0435\u043e\u0440\u0435\u043c\u043e\u0439 \u0441\u0438\u043d\u0443\u0441\u043e\u0432: \u0441\u0442\u043e\u0440\u043e\u043d\u044b \u0442\u0440\u0435\u0443\u0433\u043e\u043b\u044c\u043d\u0438\u043a\u0430 \u043f\u0440\u043e\u043f\u043e\u0440\u0446\u0438\u043e\u043d\u0430\u043b\u044c\u043d\u044b \u0441\u0438\u043d\u0443\u0441\u0430\u043c \u043f\u0440\u043e\u0442\u0438\u0432\u043e\u043b\u0435\u0436\u0430\u0449\u0438\u0445 \u0443\u0433\u043b\u043e\u0432. <\/p>\n\n\n\n

\\(\\frac{AC}{\\sin (\\measuredangle AOC)}=\\frac{AO}{\\sin (\\measuredangle OCA)}\\)<\/p>\n\n\n\n

\u041f\u043e\u0434\u0441\u0442\u0430\u0432\u0438\u043c \u043d\u0430\u0439\u0434\u0435\u043d\u043d\u044b\u0435 \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u044f \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u044f \u0443\u0433\u043b\u043e\u0432:<\/p>\n\n\n\n

\\(\\frac{2R}{\\sin (135^{\\circ})}=\\frac{AO}{\\sin (45^{\\circ}-\\frac{\\alpha}{2})}\\)<\/p>\n\n\n\n

\\(AO=\\frac{2R}{\\frac{\\sqrt{2}}{2}}\\cdot \\sin (45^{\\circ}-\\frac{\\alpha}{2})=2\\sqrt{2}R\\cdot \\sin (45^{\\circ}-\\frac{\\alpha}{2})\\) (3)<\/p>\n\n\n\n

\u041e\u0442\u0432\u0435\u0442: \\(AO=2\\sqrt{2}R\\cdot \\sin (45^{\\circ}-\\frac{\\alpha}{2})\\)<\/p>\n\n\n\n

(\u05d1)<\/mark><\/strong> \u041d\u0430\u0439\u0434\u0435\u043c \u0432\u0435\u043b\u0438\u0447\u0438\u043d\u0443 \u0443\u0433\u043b\u0430 \\(\\alpha\\) \u043f\u0440\u0438 \u0443\u0441\u043b\u043e\u0432\u0438\u0438, \u0447\u0442\u043e \\(AO=R\\sqrt{2}\\). \u041f\u043e\u0434\u0441\u0442\u0430\u0432\u0438\u043c \u0437\u043d\u0430\u0447\u0435\u043d\u0438\u0435 \\(AO=R\\sqrt{2}\\) \u0432 \u0440\u0430\u0432\u0435\u043d\u0441\u0442\u0432\u043e (3)<\/p>\n\n\n\n

\\(R\\sqrt{2}=2\\sqrt{2}R\\cdot \\sin (45^{\\circ}-\\frac{\\alpha}{2})\\)<\/p>\n\n\n\n

\u0420\u0430\u0437\u0434\u0435\u043b\u0438\u043c \u043e\u0431\u0435 \u0447\u0430\u0441\u0442\u0438 \u0440\u0430\u0432\u0435\u043d\u0441\u0442\u0432\u0430 \u043d\u0430 \\(R\\sqrt{2}\\). \u041f\u043e\u043b\u0443\u0447\u0438\u043c:<\/p>\n\n\n\n

\\(2 \\sin (45^{\\circ}-\\frac{\\alpha}{2})=1\\)<\/p>\n\n\n\n

\\(\\sin (45^{\\circ}-\\frac{\\alpha}{2})=\\frac{1}{2}\\)<\/p>\n\n\n\n

\\(45^{\\circ}-\\frac{\\alpha}{2}=30^{\\circ}\\)<\/p>\n\n\n\n

\\(\\frac{\\alpha}{2}=15^{\\circ}\\), \\(\\alpha=30^{\\circ}\\).<\/p>\n\n\n\n

\u041e\u0442\u0432\u0435\u0442: \\(\\alpha=30^{\\circ}\\).<\/p>\n\n\n\n

(\u05d2)<\/mark><\/strong> \u041d\u0430\u0439\u0434\u0435\u043c \\(R\\) \u043f\u0440\u0438 \u0443\u0441\u043b\u043e\u0432\u0438\u0438, \u0447\u0442\u043e \u043f\u043b\u043e\u0449\u0430\u0434\u044c \u0434\u0435\u043b\u044c\u0442\u043e\u0438\u0434\u0430 \u0440\u0430\u0432\u043d\u0430 \\(25\\sqrt{3}\\). <\/p>\n\n\n\n

\u0422\u0430\u043a \u043a\u0430\u043a \u0434\u0438\u0430\u0433\u043e\u043d\u0430\u043b\u0438 \u0434\u0435\u043b\u044c\u0442\u043e\u0438\u0434\u0430 \u043f\u0435\u0440\u043f\u0435\u043d\u0434\u0438\u043a\u0443\u043b\u044f\u0440\u043d\u044b, \u0435\u0433\u043e \u043f\u043b\u043e\u0449\u0430\u0434\u044c \u0440\u0430\u0432\u043d\u0430 \u043f\u043e\u043b\u043e\u0432\u0438\u043d\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u0435\u0434\u0435\u043d\u0438\u044f \u0434\u0438\u0430\u0433\u043e\u043d\u0430\u043b\u0435\u0439. <\/p>\n\n\n\n

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

\u0418\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 \\(ABC\\) \u043d\u0430\u0439\u0434\u0435\u043c \\(AB\\):<\/p>\n\n\n\n

\\(AB=AC\\cdot \\cos (\\measuredangle CAB)=2R\\cos 30^{\\circ}=2R\\cdot \\frac{\\sqrt{3}}{2}=\\)<\/p>\n\n\n\n

\\(=R\\sqrt{3}\\)<\/p>\n\n\n\n

\u0418\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 \\(ABK\\) \u043d\u0430\u0439\u0434\u0435\u043c \\(BK\\):<\/p>\n\n\n\n

\\(BK=AB\\cdot \\sin (\\measuredangle KAB)=R\\sqrt{3} \\cdot \\sin 30^{\\circ}=\\frac{R\\sqrt{3}}{2}\\)<\/p>\n\n\n\n

\\(DB=2BK=R\\sqrt{3}\\)<\/p>\n\n\n\n

\\(S=\\frac{AC\\cdot DB}{2}=\\frac{2R\\cdot R\\sqrt{3}}{2}=\\frac{2R^2\\sqrt{3}}{2}=R^2\\sqrt{3}\\)<\/p>\n\n\n\n

\\(R^2\\sqrt{3}=25\\sqrt{3}\\). \u041e\u0442\u0441\u044e\u0434\u0430 \\(R=5\\)<\/p>\n\n\n\n

\u041e\u0442\u0432\u0435\u0442: \\(R=5\\)<\/p>\n\n\n\n

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

\u0417\u0430\u043c\u0435\u0442\u0438\u043c, \u0447\u0442\u043e \u0432 \u0442\u0440\u0435\u0443\u0433\u043e\u043b\u044c\u043d\u0438\u043a\u0435 \\(ABD\\) \\(AD=AB\\), \\(\\measuredangle DAC=\\measuredangle DAC=30^{\\circ}\\). \u0422\u043e \u0435\u0441\u0442\u044c \\(\\measuredangle DAB=60^{\\circ}\\), \u0441\u043b\u0435\u0434\u043e\u0432\u0430\u0442\u0435\u043b\u044c\u043d\u043e \u0442\u0440\u0435\u0443\u0433\u043e\u043b\u044c\u043d\u0438\u043a \\(ABD\\) — \u0440\u0430\u0432\u043d\u043e\u0441\u0442\u043e\u0440\u043e\u043d\u043d\u0438\u0439 \u0438 \\(DB=AB=R\\sqrt{3}\\).<\/p>\n\n\n\n

\n
\n

(\u05d3)<\/mark><\/strong> \u041d\u0430\u0439\u0434\u0435\u043c \u0434\u043b\u0438\u043d\u0443 \u043e\u0442\u0440\u0435\u0437\u043a\u0430 \\(O_1O\\). \u0420\u0430\u0441\u0441\u043c\u043e\u0442\u0440\u0438\u043c \u0442\u0440\u0435\u0443\u0433\u043e\u043b\u044c\u043d\u0438\u043a \\(AO_1O\\). <\/p>\n\n\n\n

\\(AO_1=R=5\\), <\/p>\n\n\n\n

\\( \\measuredangle O_1AO=\\frac{\\alpha}{2}=15^{\\circ}\\), <\/p>\n\n\n\n

\\(AO=2\\sqrt{2}R\\cdot \\sin (45^{\\circ}-\\frac{\\alpha}{2})=\\) [\u0438\u0437 (\u05d0)<\/strong> (2)]<\/p>\n\n\n\n

\\(=2\\sqrt{2}\\cdot 5 \\cdot \\sin (45^{\\circ}-15^{\\circ})=5\\sqrt{2}\\).<\/p>\n\n\n\n

\u0412\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:<\/p>\n\n\n\n

\\(O_1O^2=AO_1^2+AO^2-2\\cdot AO_1\\cdot AO \\cdot \\cos (\\measuredangle O_1AO)\\)<\/p>\n\n\n\n

\\(O_1O^2=5^2+(5\\sqrt{2})^2-2\\cdot 5\\cdot 5\\sqrt{2}\\cdot \\cos (15^{\\circ})=\\)<\/p>\n\n\n\n

\\(=75-5\\sqrt{2}\\cos \\frac{\\pi}{12}\\approx 4.29\\)<\/p>\n\n\n\n

\\(O_1O=\\sqrt {4,29}\\approx 2.07\\)<\/p>\n\n\n\n

\u041e\u0442\u0432\u0435\u0442:<\/em> 2.07<\/p>\n\n\n\n

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

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

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

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

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