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Консультация № 195803
04.06.2019, 17:35
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Здравствуйте! Прошу помощи в следующем вопросе:
В выпуклом равностороннем пятиугольнике ABCDE угол ABC равен 136 градусам, угол BCD равен 104 градусам. Найти угол DEA в градусах.
В выпуклом равностороннем пятиугольнике ABCDE угол ABC равен 136 градусам, угол BCD равен 104 градусам. Найти угол DEA в градусах.
Обсуждение
08.06.2019, 06:26
общий
Адресаты:
Сообщите, пожалуйста, из какого источника Вы взяли эту задачу.
Об авторе:
Facta loquuntur.
Facta loquuntur.
08.06.2019, 19:16
общий
это ответ
Здравствуйте, karmapolozhitelnaya!
Согласно свойству равнобедренного треугольника [1, с. 31] и теореме о сумме углов треугольника [1, с. 42], [$8736$]BCA=(180[$186$]-136[$186$])/2=22[$186$]. Тогда [$8736$]ACD=104[$186$]-22[$186$]=82[$186$].
Пусть a -- длина стороны рассматриваемого пятиугольника. По теореме косинусов [1, c. 148] с учётом формулы на с. 110 [2] получим
С другой стороны,
Тогда
Литература
1. Погорелов А. В. Геометрия: учебное пособие для 6 -- 10 классов средней школы. -- М.: Просвещение, 1986. -- 304 с.
2. Цыпкин А. Г., Цыпкин Г. Г. Математические формулы. -- М.: Наука, 1985. -- 128 с.
Согласно свойству равнобедренного треугольника [1, с. 31] и теореме о сумме углов треугольника [1, с. 42], [$8736$]BCA=(180[$186$]-136[$186$])/2=22[$186$]. Тогда [$8736$]ACD=104[$186$]-22[$186$]=82[$186$].
Пусть a -- длина стороны рассматриваемого пятиугольника. По теореме косинусов [1, c. 148] с учётом формулы на с. 110 [2] получим
|AC|2=a2+a2-2a2cos136[$186$]=2a2(1-cos136[$186$])=4a2sin268[$186$], |AC|=2a*sin68[$186$];
|AD|2=|AC|2+a2-2|AC|a*cos82[$186$]=a2(1+4*sin68[$186$]*(sin68[$186$]-cos82[$186$])).
С другой стороны,
|AD|2=a2+a2-2a2cos[$8736$]DEA=2a2(1-cos[$8736$]DEA).
Тогда
a2(1+4*sin68[$186$]*(sin68[$186$]-cos82[$186$]))=2a2(1-cos[$8736$]DEA),
1+4*sin68[$186$]*(sin68[$186$]-cos82[$186$])=2-2*cos[$8736$]DEA [2, с. 108],
4*sin68[$186$]*2*cos38[$186$]*sin30[$186$]=1-2*cos[$8736$]DEA,
4*sin68[$186$]*cos38[$186$]=1-2*cos[$8736$]DEA,
cos[$8736$]DEA=(1-4*sin68[$186$]*cos38[$186$])/2=(1-2(sin30[$186$]+sin106[$186$]))/2=(1-2(1/2+sin106[$186$]))/2=-sin106[$186$]=sin(-106[$186$])=sin(-74[$186$]),
[$8736$]DEA=arccos(sin(-74[$186$]))=90[$186$]-arcsin(sin(-74[$186$]))=90[$186$]+74[$186$]=164[$186$] [2, с. 111].
Литература
1. Погорелов А. В. Геометрия: учебное пособие для 6 -- 10 классов средней школы. -- М.: Просвещение, 1986. -- 304 с.
2. Цыпкин А. Г., Цыпкин Г. Г. Математические формулы. -- М.: Наука, 1985. -- 128 с.
5
Об авторе:
Facta loquuntur.
Facta loquuntur.
Форма ответа
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