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Консультация № 136838
14.05.2008, 20:03
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Дорогие Эксперты требуется помощь в вопросе о:
Доказать что отрезок соединяющий середины диагоналей трапеции параллен ее основаниям и равен их полуразности.
Доказать что отрезок соединяющий середины диагоналей трапеции параллен ее основаниям и равен их полуразности.
Обсуждение
15.05.2008, 02:26
общий
это ответ
Здравствуйте, Coolcooler1!
Изобразите трапецию ABCD, у которой AD=a - нижнее основание, BC=b - верхнее основание. Обозначьте через E середину диагонали AC, через F - середину диагонали BD. Пусть AD>BC.
Из теоремы Фалеса (и из того, что средняя линия трапеции параллельна основаниям,) следует, что если G - середина стороны AB, а H - середина стороны CD, то отрезок GH - средняя линия трапеции - пересечет диагонали трапеции как раз в точках E и F. Следовательно, отрезок, соединяющий середины диагоналей трапеции, параллелен ее основаниям.
Далее, EF=GF-GE=AD/2-BC/2=(AD-BC)/2. Следовательно, отрезок, соединяющий середины диагоналей трапеции, равен их полуразности.
Было приятно вспомнить школьные годы...
С уважением.
Изобразите трапецию ABCD, у которой AD=a - нижнее основание, BC=b - верхнее основание. Обозначьте через E середину диагонали AC, через F - середину диагонали BD. Пусть AD>BC.
Из теоремы Фалеса (и из того, что средняя линия трапеции параллельна основаниям,) следует, что если G - середина стороны AB, а H - середина стороны CD, то отрезок GH - средняя линия трапеции - пересечет диагонали трапеции как раз в точках E и F. Следовательно, отрезок, соединяющий середины диагоналей трапеции, параллелен ее основаниям.
Далее, EF=GF-GE=AD/2-BC/2=(AD-BC)/2. Следовательно, отрезок, соединяющий середины диагоналей трапеции, равен их полуразности.
Было приятно вспомнить школьные годы...
С уважением.
Об авторе:
Facta loquuntur.
Facta loquuntur.
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