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МатематикаГДЗ по геометрии 7 класс Атанасян Задание 765
Задание 765
\[\boxed{\mathbf{765.ОК\ ГДЗ - домашка\ на}\ 5}\]
\[Дано:\]
\[X,Y,Z - произвольные\ точки.\]
\[Доказать:\ \]
\[\overrightarrow{p} = \overrightarrow{\text{XY}} + \overrightarrow{\text{ZX}} + \overrightarrow{\text{YZ}} = \overrightarrow{0};\]
\[\overrightarrow{q} = \left( \overrightarrow{\text{XY}} - \overrightarrow{\text{XZ}} \right) + \overrightarrow{\text{YZ}} = \overrightarrow{0};\]
\[\overrightarrow{z} = \left( \overrightarrow{\text{ZY}} - \overrightarrow{\text{XY}} \right) - \overrightarrow{\text{ZX}} = \overrightarrow{0}.\]
\[Доказательство.\]
\[Воспользуемся\ правилом\ \]
\[многоугольника.\]
\[1)\ \overrightarrow{p} = \overrightarrow{\text{XY}} + \overrightarrow{\text{ZX}} + \overrightarrow{\text{YZ}} =\]
\[= \overrightarrow{\text{XY}} + \overrightarrow{\text{YZ}} + \overrightarrow{\text{ZX}} = \overrightarrow{\text{XX}} = \overrightarrow{0}.\]
\[2)\ \overrightarrow{q} = \left( \overrightarrow{\text{XY}} - \overrightarrow{\text{XZ}} \right) + \overrightarrow{\text{YZ}} =\]
\[= \left( \overrightarrow{\text{XY}} + \overrightarrow{\text{ZX}} \right) + \overrightarrow{\text{YZ}} =\]
\[= \left( \overrightarrow{\text{XY}} + \overrightarrow{\text{YZ}} \right) + \overrightarrow{\text{ZX}} =\]
\[= \overrightarrow{\text{XZ}} + \overrightarrow{\text{ZX}} = \overrightarrow{\text{XX}} = \overrightarrow{0}.\]
\[3)\ \overrightarrow{z} = \left( \overrightarrow{\text{ZY}} - \overrightarrow{\text{XY}} \right) - \overrightarrow{\text{ZX}} =\]
\[= \left( \overrightarrow{\text{ZY}} + \overrightarrow{\text{YX}} \right) + \overrightarrow{\text{XZ}} = \overrightarrow{\text{ZX}} + \overrightarrow{\text{XZ}} =\]
\[= \overrightarrow{\text{ZZ}} = \overrightarrow{0}.\]
\[Таким\ образом:\ \]
\[\overrightarrow{p} = \overrightarrow{q} = \overrightarrow{z} = \overrightarrow{0}.\]
\[Что\ и\ требовалось\ доказать\text{.\ }\]
