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>A 平行4辺形の条件

平行4辺形の条件

平行4辺形になる条件は、5つある。


平行4辺形の条件 1:2組の対辺が、それぞれ平行である。

\begin{xy} {(0,-15) \ar @{->}(15,-15)}, {(15,-15) \ar @{-}(30,-15)}, {(10,0) \ar @{->}(25,0)}, {(25,0) \ar @{-}(40,0)}, {(0,-15) \ar @{->}(5,-7)}, {(5,-7) \ar @{-}(10,0)}, {(30,-15) \ar @{->}(35,-7)}, {(35,-7) \ar @{-}(40,0)}, \end{xy}


平行4辺形の条件 2:2組の対辺が、それぞれ等しい。

\begin{xy} {(0,-15) \ar @{-|}(15,-15)}, {(15,-15) \ar @{-}(30,-15)}, {(10,0) \ar @{-|}(25,0)}, {(25,0) \ar @{-}(40,0)}, {(0,-15) \ar @{-||}(5,-7)}, {(5,-7) \ar @{-}(10,0)}, {(30,-15) \ar @{-||}(35,-7)}, {(35,-7) \ar @{-}(40,0)}, \end{xy}


\begin{xy} {(0,-15) \ar @{->}(10,-15)}, {(10,-15) \ar @{-|}(15,-15)}, {(15,-15) \ar @{-}(30,-15)}, {(10,0) \ar @{->}(20,0)}, {(20,0) \ar @{-|}(25,0)}, {(25,0) \ar @{-}(40,0)}, {(0,-15) \ar @{-}(5,-7)}, {(5,-7) \ar @{-}(10,0)}, {(30,-15) \ar @{-}(35,-7)}, {(35,-7) \ar @{-}(40,0)}, \end{xy}


平行4辺形の条件 4:2組の対角が、それぞれ等しい。

\begin{xy} {(0,-15) \ar @{-}(30,-15)}, {(10,0) \ar @{-}(40,0)}, {(0,-15) \ar @{-}(10,0)}, {(30,-15) \ar @{-}(40,0)}, (11,-2)*{\scriptsize 〇}="A", (29,-13)*{\scriptsize 〇}="A", (4,-13)*{\small ●}="A", (36,-2)*{\small ●}="A", \end{xy}


平行4辺形の条件 5:対角線が、それぞれの中点で交わる。

\begin{xy} {(0,-15) \ar @{-}(30,-15)}, {(10,0) \ar @{-}(40,0)}, {(0,-15) \ar @{-}(10,0)}, {(30,-15) \ar @{-}(40,0)}, {(0,-15) \ar @{-|}(8,-12)}, {(8,-12) \ar @{-|}(32,-3)}, {(32,-3) \ar @{-}(40,0)}, {(10,0) \ar @{-||}(14,-3)}, {(14,-3) \ar @{-||}(26,-12)}, {(26,-12) \ar @{-}(30,-15)}, \end{xy}


図形が、平行4辺形かどうかを判断するためには、対辺、対角、対角線、に注目するとよい。



<例題 \( \Large 1 \) >以下から平行4辺形を見つけなさい。また、平行4辺形になる条件も、書きなさい。

\begin{xy} (10,3)*{\small A}="A", (40,3)*{\small B}="A", (0,-18)*{\small D}="A", (30,-18)*{\small C}="A", {(0,-15) \ar @{->}(15,-15)}, {(15,-15) \ar @{-}(30,-15)}, {(10,0) \ar @{->}(25,0)}, {(25,0) \ar @{-}(40,0)}, {(0,-15) \ar @{->}(5,-7)}, {(5,-7) \ar @{-}(10,0)}, {(30,-15) \ar @{->}(35,-7)}, {(35,-7) \ar @{-}(40,0)}, \end{xy} \begin{xy} (10,3)*{\small E}="A", (40,3)*{\small F}="A", (0,-18)*{\small H}="A", (30,-18)*{\small G}="A", (15,-18)*{\small 5}="A", (25,3)*{\small 5}="A", {(0,-15) \ar @{-}(15,-15)}, {(15,-15) \ar @{-}(30,-15)}, {(10,0) \ar @{-}(25,0)}, {(25,0) \ar @{-}(40,0)}, {(0,-15) \ar @{->}(5,-7)}, {(5,-7) \ar @{-}(10,0)}, {(30,-15) \ar @{->}(35,-7)}, {(35,-7) \ar @{-}(40,0)}, \end{xy} \begin{xy} (10,3)*{\small I}="A", (40,3)*{\small J}="A", (0,-18)*{\small L}="A", (30,-18)*{\small K}="A", (15,-18)*{\small 5}="A", (25,3)*{\small 5}="A", {(0,-15) \ar @{->}(15,-15)}, {(15,-15) \ar @{-}(30,-15)}, {(10,0) \ar @{->}(25,0)}, {(25,0) \ar @{-}(40,0)}, {(0,-15) \ar @{-}(5,-7)}, {(5,-7) \ar @{-}(10,0)}, {(30,-15) \ar @{-}(35,-7)}, {(35,-7) \ar @{-}(40,0)}, \end{xy} \begin{xy} (10,3)*{\small M}="A", (40,3)*{\small N}="A", (0,-18)*{\small P}="A", (30,-18)*{\small O}="A", (15,-18)*{\small 5}="A", (25,3)*{\small 5}="A", (2,-8)*{\small 3}="A", (37,-8)*{\small 3}="A", {(0,-15) \ar @{-}(15,-15)}, {(15,-15) \ar @{-}(30,-15)}, {(10,0) \ar @{-}(25,0)}, {(25,0) \ar @{-}(40,0)}, {(0,-15) \ar @{-}(5,-7)}, {(5,-7) \ar @{-}(10,0)}, {(30,-15) \ar @{-}(35,-7)}, {(35,-7) \ar @{-}(40,0)}, \end{xy} \begin{xy} (10,3)*{\small Q}="A", (40,3)*{\small R}="A", (0,-18)*{\small T}="A", (30,-18)*{\small S}="A", (2,-8)*{\small 3}="A", (37,-8)*{\small 3}="A", {(0,-15) \ar @{-}(15,-15)}, {(15,-15) \ar @{-}(30,-15)}, {(10,0) \ar @{-}(25,0)}, {(25,0) \ar @{-}(40,0)}, {(0,-15) \ar @{-}(5,-7)}, {(5,-7) \ar @{-}(10,0)}, {(30,-15) \ar @{-}(35,-7)}, {(35,-7) \ar @{-}(40,0)}, \end{xy} \begin{xy} (10,3)*{\small U}="A", (40,3)*{\small V}="A", (0,-18)*{\small X}="A", (30,-18)*{\small W}="A", (8,-10)*{\small 3}="A", (31,-6)*{\small 3}="A", (17,-3)*{\small 2}="A", (23,-13)*{\small 2}="A", {(0,-15) \ar @{-}(30,-15)}, {(10,0) \ar @{-}(40,0)}, {(0,-15) \ar @{-}(10,0)}, {(30,-15) \ar @{-}(40,0)}, {(0,-15) \ar @{-}(8,-12)}, {(8,-12) \ar @{-}(32,-3)}, {(32,-3) \ar @{-}(40,0)}, {(10,0) \ar @{-}(14,-3)}, {(14,-3) \ar @{-}(26,-12)}, {(26,-12) \ar @{-}(30,-15)}, \end{xy}


<解答 \( \Large 1 \) >
\( \square ABCD \)
平行4辺形の条件:2組の対辺が、それぞれ平行である。

\( \square IJKL \)
平行4辺形の条件:1組の対辺が、平行で、長さが等しい。

\( \square MNOP \)
平行4辺形の条件:2組の対辺が、それぞれ等しい。

\( \square UVWX \)
平行4辺形の条件:対角線が、それぞれの中点で交わる。

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