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辺と面の関係
立体図形では、辺と面は、3つの関係になれる。交わるか、平行か、含まれるかだ。
例えば、直方体ABCD-EFGHで、辺と面の関係を、考えていこう。
\begin{xy}
(-3,27)*{A}="A",
(23,15)*{B}="A",
(-23,15)*{D}="A",
(-3,-3)*{C}="A",
(-3,9)*{E}="A",
(23,-9)*{F}="A",
(-23,-9)*{H}="A",
(-3,-21)*{G}="A",
(0,0)*{\bullet}="A",
(20,12)*{\bullet}="A",
(-20,12)*{\bullet}="A",
(20,-6)*{\bullet}="A",
(-20,-6)*{\bullet}="A",
(0,-18)*{\bullet}="A",
(0,6)*{\bullet}="A",
(0,24)*{\bullet}="A",
{(0,0) \ar @{-}(20,12)},
{(0,0) \ar @{-}(0,-18)},
{(0,0) \ar @{-}(-20,12)},
{(0,24) \ar @{-}(20,12)},
{(0,24) \ar @{-}(-20,12)},
{(20,12) \ar @{-}(20,-6)},
{(-20,12) \ar @{-}(-20,-6)},
{(0,-18) \ar @{-}(20,-6)},
{(0,-18) \ar @{-}(-20,-6)},
{(0,6) \ar @{.}(20,-6)},
{(0,6) \ar @{.}(-20,-6)},
{(0,6) \ar @{.}(0,24)},
\end{xy}
辺と面が交わる
辺ABが交わるのは、面辺ADHEと面BCGHだ。
交わるとは、1点を共有する、ともいえる。
\begin{xy}
(-3,27)*{A}="A",
(23,15)*{B}="A",
(-23,15)*{D}="A",
(-3,-3)*{C}="A",
(-3,9)*{E}="A",
(23,-9)*{F}="A",
(-23,-9)*{H}="A",
(-3,-21)*{G}="A",
(0,0)*{\bullet}="A",
(20,12)*{\bullet}="A",
(-20,12)*{\bullet}="A",
(20,-6)*{\bullet}="A",
(-20,-6)*{\bullet}="A",
(0,-18)*{\bullet}="A",
(0,6)*{\bullet}="A",
(0,24)*{\bullet}="A",
{(0,0) \ar @{-}(20,12)},
{(0,0) \ar @{-}(0,-18)},
{(0,24) \ar @{-}(20,12)},
{(0,24) \ar @{-}(-20,12)},
{(20,12) \ar @{-}(20,-6)},
{(-20,12) \ar @{-}(-20,-6)},
{(0,-18) \ar @{-}(20,-6)},
{(0,6) \ar @{.}(-20,-6)},
{(0,6) \ar @{.}(0,24)},
\end{xy}
<例題 \( \Large 1 \) >直方体ABCD-EFGHで、辺BFと交わる面を、すべて書きなさい。
\begin{xy}
(-3,27)*{A}="A",
(23,15)*{B}="A",
(-23,15)*{D}="A",
(-3,-3)*{C}="A",
(-3,9)*{E}="A",
(23,-9)*{F}="A",
(-23,-9)*{H}="A",
(-3,-21)*{G}="A",
(0,0)*{\bullet}="A",
(20,12)*{\bullet}="A",
(-20,12)*{\bullet}="A",
(20,-6)*{\bullet}="A",
(-20,-6)*{\bullet}="A",
(0,-18)*{\bullet}="A",
(0,6)*{\bullet}="A",
(0,24)*{\bullet}="A",
{(0,0) \ar @{-}(20,12)},
{(0,0) \ar @{-}(0,-18)},
{(0,0) \ar @{-}(-20,12)},
{(0,24) \ar @{-}(20,12)},
{(0,24) \ar @{-}(-20,12)},
{(20,12) \ar @{-}(20,-6)},
{(-20,12) \ar @{-}(-20,-6)},
{(0,-18) \ar @{-}(20,-6)},
{(0,-18) \ar @{-}(-20,-6)},
{(0,6) \ar @{.}(20,-6)},
{(0,6) \ar @{.}(-20,-6)},
{(0,6) \ar @{.}(0,24)},
\end{xy}
辺と面が平行
辺ABが平行なのは、面辺CDHGと面EFGHだ。
平行な辺と平行な面は、どこまで延長しても、交わらない。
\begin{xy}
(-3,27)*{A}="A",
(23,15)*{B}="A",
(-23,15)*{D}="A",
(-3,-3)*{C}="A",
(-3,9)*{E}="A",
(23,-9)*{F}="A",
(-23,-9)*{H}="A",
(-3,-21)*{G}="A",
(0,0)*{\bullet}="A",
(20,12)*{\bullet}="A",
(-20,12)*{\bullet}="A",
(20,-6)*{\bullet}="A",
(-20,-6)*{\bullet}="A",
(0,-18)*{\bullet}="A",
(0,6)*{\bullet}="A",
(0,24)*{\bullet}="A",
{(0,0) \ar @{-}(-20,12)},
{(0,0) \ar @{-}(0,-18)},
{(0,24) \ar @{-}(20,12)},
{(-20,12) \ar @{-}(-20,-6)},
{(-20,-6) \ar @{-}(0,-18)},
{(-20,-6) \ar @{.}(0,6)},
{(0,6) \ar @{.}(20,-6)},
{(0,-18) \ar @{-}(20,-6)},
\end{xy}
<例題 \( \Large 2 \) >直方体ABCD-EFGHで、辺BFと平行な面を、すべて書きなさい。
\begin{xy}
(-3,27)*{A}="A",
(23,15)*{B}="A",
(-23,15)*{D}="A",
(-3,-3)*{C}="A",
(-3,9)*{E}="A",
(23,-9)*{F}="A",
(-23,-9)*{H}="A",
(-3,-21)*{G}="A",
(0,0)*{\bullet}="A",
(20,12)*{\bullet}="A",
(-20,12)*{\bullet}="A",
(20,-6)*{\bullet}="A",
(-20,-6)*{\bullet}="A",
(0,-18)*{\bullet}="A",
(0,6)*{\bullet}="A",
(0,24)*{\bullet}="A",
{(0,0) \ar @{-}(20,12)},
{(0,0) \ar @{-}(0,-18)},
{(0,0) \ar @{-}(-20,12)},
{(0,24) \ar @{-}(20,12)},
{(0,24) \ar @{-}(-20,12)},
{(20,12) \ar @{-}(20,-6)},
{(-20,12) \ar @{-}(-20,-6)},
{(0,-18) \ar @{-}(20,-6)},
{(0,-18) \ar @{-}(-20,-6)},
{(0,6) \ar @{.}(20,-6)},
{(0,6) \ar @{.}(-20,-6)},
{(0,6) \ar @{.}(0,24)},
\end{xy}
辺が面に含まれる
辺ABが含まれるのは、面辺ABCDと面ABFEだ。
\begin{xy}
(-3,27)*{A}="A",
(23,15)*{B}="A",
(-23,15)*{D}="A",
(-3,-3)*{C}="A",
(-3,9)*{E}="A",
(23,-9)*{F}="A",
(0,0)*{\bullet}="A",
(20,12)*{\bullet}="A",
(-20,12)*{\bullet}="A",
(20,-6)*{\bullet}="A",
(0,6)*{\bullet}="A",
(0,24)*{\bullet}="A",
{(0,0) \ar @{-}(20,12)},
{(0,0) \ar @{-}(-20,12)},
{(0,24) \ar @{-}(20,12)},
{(0,24) \ar @{-}(-20,12)},
{(20,12) \ar @{-}(20,-6)},
{(0,6) \ar @{.}(20,-6)},
{(0,6) \ar @{.}(0,24)},
\end{xy}
<例題 \( \Large 3 \) >直方体ABCD-EFGHで、辺BFが含まれる面を、すべて書きなさい。
\begin{xy}
(-3,27)*{A}="A",
(23,15)*{B}="A",
(-23,15)*{D}="A",
(-3,-3)*{C}="A",
(-3,9)*{E}="A",
(23,-9)*{F}="A",
(-23,-9)*{H}="A",
(-3,-21)*{G}="A",
(0,0)*{\bullet}="A",
(20,12)*{\bullet}="A",
(-20,12)*{\bullet}="A",
(20,-6)*{\bullet}="A",
(-20,-6)*{\bullet}="A",
(0,-18)*{\bullet}="A",
(0,6)*{\bullet}="A",
(0,24)*{\bullet}="A",
{(0,0) \ar @{-}(20,12)},
{(0,0) \ar @{-}(0,-18)},
{(0,0) \ar @{-}(-20,12)},
{(0,24) \ar @{-}(20,12)},
{(0,24) \ar @{-}(-20,12)},
{(20,12) \ar @{-}(20,-6)},
{(-20,12) \ar @{-}(-20,-6)},
{(0,-18) \ar @{-}(20,-6)},
{(0,-18) \ar @{-}(-20,-6)},
{(0,6) \ar @{.}(20,-6)},
{(0,6) \ar @{.}(-20,-6)},
{(0,6) \ar @{.}(0,24)},
\end{xy}
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