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Консультация № 189769
13.09.2016, 18:53
0.00 руб.
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Здравствуйте, уважаемые эксперты! Если вам не составит труда,прошу вас ответить на следующий вопрос:
Прикрепленные файлы:
Обсуждение
14.09.2016, 23:07
общий
это ответ
Здравствуйте, moonfox!
a)
Середины диагоналей лежат на серединах высот трапеции, поэтому PQ||AD||BC
отсюда следует и подобие треугольников [$8895$]AOD, [$8895$]POQ и [$8895$]BOC
Учитывая, что P - середина AC
OP=OA-(1/2)(OA+OC)
при этом, из подобия треугольников следует
PQ/OP=AD/OA=BC/OC
что даёт нам
PQ=AD-(1/2)(AD+BC)=a-(a+b)/2=(a-b)/2
Гипотенуза прямоугольного треугольника является диаметром описанной окружности, поэтому опущенная на гипотенузу медиана равна половине гипотенузы.
OM=AD/2=a/2
ON=BC/2=b/2
MN=(a+b)/2
В итоге
MN[$183$]PQ=(a+b)(a-b)/4=(a2-b2)/4
б)
Проведём CF||AB
При этом отсекаются параллелограммы, в которых MP=AF=BC
С другой стороны, из подобия треугольников [$916$]CFD и [$916$]CPN
PN/FD=CN/CD=3/4
Итого, MN=MP+PN=BC+(3/4)(AD-BC)=4
a)
Середины диагоналей лежат на серединах высот трапеции, поэтому PQ||AD||BC
отсюда следует и подобие треугольников [$8895$]AOD, [$8895$]POQ и [$8895$]BOC
Учитывая, что P - середина AC
OP=OA-(1/2)(OA+OC)
при этом, из подобия треугольников следует
PQ/OP=AD/OA=BC/OC
что даёт нам
PQ=AD-(1/2)(AD+BC)=a-(a+b)/2=(a-b)/2
Гипотенуза прямоугольного треугольника является диаметром описанной окружности, поэтому опущенная на гипотенузу медиана равна половине гипотенузы.
OM=AD/2=a/2
ON=BC/2=b/2
MN=(a+b)/2
В итоге
MN[$183$]PQ=(a+b)(a-b)/4=(a2-b2)/4
б)
Проведём CF||AB
При этом отсекаются параллелограммы, в которых MP=AF=BC
С другой стороны, из подобия треугольников [$916$]CFD и [$916$]CPN
PN/FD=CN/CD=3/4
Итого, MN=MP+PN=BC+(3/4)(AD-BC)=4
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