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Консультация № 144003
15.09.2008, 17:18
0.00 руб.
15.09.2008, 17:19
0
1
1
В выпуклом четырёхугольнике KLMN точки Е, F, G, Н являются соответственно серединами сторон KL, LM, MN, NK. Площадь четырёхугольника EFGH равна Q, угол HEF равен 30 градусов , угол EFH равен 90 градусов . Найти длины диагоналей четырёхугольника.
Обсуждение
16.09.2008, 12:00
общий
это ответ
Здравствуйте, Олег Валерьевич!
Заметим, что EFGH - параллелограмм. Действительно, FG - средняя линия треугольника LMN и потому параллельна основанию LN. Аналогично, EH - средняя линия треугольника LNK, параллельная его основанию LN. Следовательно, FG и EH параллельны. Также убеждаемся, что параллельны EF и GH.
Рассмотрим треугольник EFH. В нём по условию угол EFH - прямой, а угол HEF равен 30 градусам. Пусть катет FH = a, тогда катет EF = a*sqrt(3), а гипотенуза EH = 2a. Площадь этого треугольника равна половине площади параллелограмма EFGH, то есть
(a^2)*sqrt(3)/2 = Q/2, и a = sqrt(Q/sqrt(3)).
Диагонали четырехугольника KLMN равны:
LN = 2*EH = 4*a = 4*sqrt(Q/sqrt(3)),
MK = 2*EF = 2*sqrt(3)*sqrt(Q/sqrt(3)) = 2*sqrt(Q*sqrt(3)).
Заметим, что EFGH - параллелограмм. Действительно, FG - средняя линия треугольника LMN и потому параллельна основанию LN. Аналогично, EH - средняя линия треугольника LNK, параллельная его основанию LN. Следовательно, FG и EH параллельны. Также убеждаемся, что параллельны EF и GH.
Рассмотрим треугольник EFH. В нём по условию угол EFH - прямой, а угол HEF равен 30 градусам. Пусть катет FH = a, тогда катет EF = a*sqrt(3), а гипотенуза EH = 2a. Площадь этого треугольника равна половине площади параллелограмма EFGH, то есть
(a^2)*sqrt(3)/2 = Q/2, и a = sqrt(Q/sqrt(3)).
Диагонали четырехугольника KLMN равны:
LN = 2*EH = 4*a = 4*sqrt(Q/sqrt(3)),
MK = 2*EF = 2*sqrt(3)*sqrt(Q/sqrt(3)) = 2*sqrt(Q*sqrt(3)).
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