<span>1) Выясняем, какое, собственно, расстояние нам нужно искать
</span><span><span>Нам нужно найти <span>расстояние между прямыми <span><span>A<span>B1</span></span><span>A<span>B1</span></span></span> и <span><span>B<span>C1</span></span><span>B<span>C1</span></span></span></span>.</span><span>Строим через прямые <span><span>A<span>B1</span></span><span>A<span>B1</span></span></span> и <span><span>B<span>C1</span></span><span>B<span>C1</span></span></span> две параллельные плоскости <span><span>A<span>B1</span><span>D1</span></span><span>A<span>B1</span><span>D1</span></span></span> и <span><span>BD<span>C1</span></span><span>BD<span>C1</span></span></span>.</span><span>Строим прямую <span><span><span>O1</span>E</span><span><span>O1</span>E</span></span>, перпендикулярную одновременно обеим плоскостям <span><span>A<span>B1</span><span>D1</span></span><span>A<span>B1</span><span>D1</span></span></span> и <span><span>BD<span>C1</span></span><span>BD<span>C1</span></span></span>.</span><span>Для дальнейшего удобства, строим прямые <span><span><span>O1</span><span>C1</span>, O<span>O1</span>, O<span>C1</span></span><span><span>O1</span><span>C1</span>, O<span>O1</span>, O<span>C1</span></span></span></span><span>Расстоянием между прямыми <span><span>A<span>B1</span></span><span>A<span>B1</span></span></span> и <span><span>B<span>C1</span></span><span>B<span>C1</span></span></span> будет расстояние <span><span><span>O1</span>E</span><span><span>O1</span>E</span></span>.
</span></span>2) Находим расстояние <span><span><span>O1</span>E</span><span><span>O1</span>E</span></span>:По свойствам куба<span><span>BD=B<span>C1</span>=D<span>C1</span>=<span>2–√</span>, <span>O1</span><span>C1</span>=<span><span>2–√</span>2</span>, O<span>C1</span>=<span><span>6–√</span>2</span>, O<span>O1</span>=1</span><span>BD=B<span>C1</span>=D<span>C1</span>=2, <span>O1</span><span>C1</span>=<span>22</span>, O<span>C1</span>=<span>62</span>, O<span>O1</span>=1</span></span><span><span><span><span>∠<span>O1</span>E<span>C1</span>=<span>90∘</span></span><span>∠<span>O1</span>E<span>C1</span>=<span>90∘</span></span></span> — по построению</span><span><span><span>∠O<span>O1</span><span>C1</span>=<span>90∘</span></span><span>∠O<span>O1</span><span>C1</span>=<span>90∘</span></span></span> — потому что прямая <span><span>O<span>O1</span></span><span>O<span>O1</span></span></span> перпендикулярна плоскости <span><span><span>A1</span><span>B1</span><span>C1</span></span><span><span>A1</span><span>B1</span><span>C1</span></span></span></span></span>Из прямоугольного треугольника <span><span>O<span>O1</span><span>C1</span></span><span>O<span>O1</span><span>C1</span></span></span><span><span>sin<span>O1</span><span>C1</span>O=<span><span>O<span>O1</span></span><span>O<span>C1</span></span></span></span><span>sin<span>O1</span><span>C1</span>O=<span><span>O<span>O1</span></span><span>O<span>C1</span></span></span></span></span>Из прямоугольного треугольника <span><span>E<span>O1</span><span>C1</span></span><span>E<span>O1</span><span>C1</span></span></span><span><span>sin<span>O1</span><span>C1</span>O=<span><span><span>O1</span>E</span><span><span>O1</span><span>C1</span></span></span></span><span>sin<span>O1</span><span>C1</span>O=<span><span><span>O1</span>E</span><span><span>O1</span><span>C1</span></span></span></span></span>Приравниваем правые части уравнений друг другу<span><span><span><span>O<span>O1</span></span><span>O<span>C1</span></span></span>=<span><span><span>O1</span>E</span><span><span>O1</span><span>C1</span></span></span></span><span><span><span>O<span>O1</span></span><span>O<span>C1</span></span></span>=<span><span><span>O1</span>E</span><span><span>O1</span><span>C1</span></span></span></span></span>Выражаем<span><span><span>O1</span>E</span><span><span>O1</span>E</span></span><span><span>O1</span>E=<span>O1</span><span>C1</span>⋅<span><span>O<span>O1</span></span><span>O<span>C1</span></span></span>=<span><span>2–√</span>2</span>⋅<span>1<span><span>3√</span><span>2√</span></span></span>=<span>1<span>3–√
</span></span></span>Ответ<span>: </span><span><span>1<span>3√</span></span><span>13</span></span><span>.</span>
</span><span><span>Нам нужно найти <span>расстояние между прямыми <span><span>A<span>B1</span></span><span>A<span>B1</span></span></span> и <span><span>B<span>C1</span></span><span>B<span>C1</span></span></span></span>.</span><span>Строим через прямые <span><span>A<span>B1</span></span><span>A<span>B1</span></span></span> и <span><span>B<span>C1</span></span><span>B<span>C1</span></span></span> две параллельные плоскости <span><span>A<span>B1</span><span>D1</span></span><span>A<span>B1</span><span>D1</span></span></span> и <span><span>BD<span>C1</span></span><span>BD<span>C1</span></span></span>.</span><span>Строим прямую <span><span><span>O1</span>E</span><span><span>O1</span>E</span></span>, перпендикулярную одновременно обеим плоскостям <span><span>A<span>B1</span><span>D1</span></span><span>A<span>B1</span><span>D1</span></span></span> и <span><span>BD<span>C1</span></span><span>BD<span>C1</span></span></span>.</span><span>Для дальнейшего удобства, строим прямые <span><span><span>O1</span><span>C1</span>, O<span>O1</span>, O<span>C1</span></span><span><span>O1</span><span>C1</span>, O<span>O1</span>, O<span>C1</span></span></span></span><span>Расстоянием между прямыми <span><span>A<span>B1</span></span><span>A<span>B1</span></span></span> и <span><span>B<span>C1</span></span><span>B<span>C1</span></span></span> будет расстояние <span><span><span>O1</span>E</span><span><span>O1</span>E</span></span>.
</span></span>2) Находим расстояние <span><span><span>O1</span>E</span><span><span>O1</span>E</span></span>:По свойствам куба<span><span>BD=B<span>C1</span>=D<span>C1</span>=<span>2–√</span>, <span>O1</span><span>C1</span>=<span><span>2–√</span>2</span>, O<span>C1</span>=<span><span>6–√</span>2</span>, O<span>O1</span>=1</span><span>BD=B<span>C1</span>=D<span>C1</span>=2, <span>O1</span><span>C1</span>=<span>22</span>, O<span>C1</span>=<span>62</span>, O<span>O1</span>=1</span></span><span><span><span><span>∠<span>O1</span>E<span>C1</span>=<span>90∘</span></span><span>∠<span>O1</span>E<span>C1</span>=<span>90∘</span></span></span> — по построению</span><span><span><span>∠O<span>O1</span><span>C1</span>=<span>90∘</span></span><span>∠O<span>O1</span><span>C1</span>=<span>90∘</span></span></span> — потому что прямая <span><span>O<span>O1</span></span><span>O<span>O1</span></span></span> перпендикулярна плоскости <span><span><span>A1</span><span>B1</span><span>C1</span></span><span><span>A1</span><span>B1</span><span>C1</span></span></span></span></span>Из прямоугольного треугольника <span><span>O<span>O1</span><span>C1</span></span><span>O<span>O1</span><span>C1</span></span></span><span><span>sin<span>O1</span><span>C1</span>O=<span><span>O<span>O1</span></span><span>O<span>C1</span></span></span></span><span>sin<span>O1</span><span>C1</span>O=<span><span>O<span>O1</span></span><span>O<span>C1</span></span></span></span></span>Из прямоугольного треугольника <span><span>E<span>O1</span><span>C1</span></span><span>E<span>O1</span><span>C1</span></span></span><span><span>sin<span>O1</span><span>C1</span>O=<span><span><span>O1</span>E</span><span><span>O1</span><span>C1</span></span></span></span><span>sin<span>O1</span><span>C1</span>O=<span><span><span>O1</span>E</span><span><span>O1</span><span>C1</span></span></span></span></span>Приравниваем правые части уравнений друг другу<span><span><span><span>O<span>O1</span></span><span>O<span>C1</span></span></span>=<span><span><span>O1</span>E</span><span><span>O1</span><span>C1</span></span></span></span><span><span><span>O<span>O1</span></span><span>O<span>C1</span></span></span>=<span><span><span>O1</span>E</span><span><span>O1</span><span>C1</span></span></span></span></span>Выражаем<span><span><span>O1</span>E</span><span><span>O1</span>E</span></span><span><span>O1</span>E=<span>O1</span><span>C1</span>⋅<span><span>O<span>O1</span></span><span>O<span>C1</span></span></span>=<span><span>2–√</span>2</span>⋅<span>1<span><span>3√</span><span>2√</span></span></span>=<span>1<span>3–√
</span></span></span>Ответ<span>: </span><span><span>1<span>3√</span></span><span>13</span></span><span>.</span>
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