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Консультация № 146357
07.10.2008, 18:03
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Средняя линия трапеции равна 10 см и делит площадь трапеции в отношении 3:5. Найти длины оснований трапеции.
Обсуждение
Неизвестный
08.10.2008, 09:31
общий
это ответ
Здравствуйте, Stepanov92!
Алгоритм решения примерно такой:
Пусть ABCD наша трапеция, LM - её средняя линия. (AD – большее основание)
Площади трапеций ALMD и LBCM равны (c учетом того что высоты трапеций равны, т.к. LM – средняя линия):
S(ALMD) = ½ AD * LM * h
S(LBCM) = ½ BC * LM * h
S(ALMD) / S(LBCM) = ½ AD * LM * h / ½ BC * LM * h = AD / ВС = 5/3
Кроме этого LM = ½ (AD + BC) cледовательно AD + BC = 2*LM
Получаем систему:
AD / ВС = 5/3
AD + BC = 2*LM (LM = 10)
Решаем:
BC = 20 – AD
3AD = 5BC = 5 * (20 – AD)
3AD = 5 * (20 – AD)
AD = 12,5
BC = 7,5
Алгоритм решения примерно такой:
Пусть ABCD наша трапеция, LM - её средняя линия. (AD – большее основание)
Площади трапеций ALMD и LBCM равны (c учетом того что высоты трапеций равны, т.к. LM – средняя линия):
S(ALMD) = ½ AD * LM * h
S(LBCM) = ½ BC * LM * h
S(ALMD) / S(LBCM) = ½ AD * LM * h / ½ BC * LM * h = AD / ВС = 5/3
Кроме этого LM = ½ (AD + BC) cледовательно AD + BC = 2*LM
Получаем систему:
AD / ВС = 5/3
AD + BC = 2*LM (LM = 10)
Решаем:
BC = 20 – AD
3AD = 5BC = 5 * (20 – AD)
3AD = 5 * (20 – AD)
AD = 12,5
BC = 7,5
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