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平行と面積の問題の解法 | 夢を叶える塾
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平行と面積の問題の解法

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平行の面積に関する性質を使った問題です。

基本問題

(1)次の図の\triangle{ADE} \equiv\triangle{CBF}を証明しなさい。
三角形の合同_4
AB//DC
AD//BC
DE//FB

(2)次の図形の\triangle{CDA}と等しい面積の三角形、\triangle{DAE}と等しい面積の三角形を答えなさい。
平行と面積_1
AB//DC

(3)次の図形でBAの延長線上に点Eを取り、\triangle{EBC}の面積が四角形ABCDの面積と等しくなるようにしたい。
どのように点Eを決めたらよいか答えなさい。
平行と面積_2

解き方

平行四辺形の性質は色々あります。
「対辺が等しい」、「対角が等しい」、「対角線が中点で交わる」等、使うシーンによって思い出せるようになっておきましょう。

また、平行線の距離が一定である性質を利用した「等積変形」は面積の問題でとても役に立ちます。

解説

(1)次の図の\triangle{ADE} \equiv\triangle{CBF}を証明しなさい。
四角形ABCDはAB//DC,AD//BCより平行四辺形です。
更に四角形EBFDもAB//DC,DE//FBより平行四辺形です。
平行四辺形の性質を使っていきます。

平行四辺形の対角は等しい性質より、
\angle{BCF}=\angle{DAE}・・・①
\angle{ABC}=\angle{CDA}・・・②
\angle{EBF}=\angle{FDE}・・・③
②③より
\angle{ABC}=\angle{CDA}
\angle{ABF}+\angle{FBC}=\angle{CDE}+\angle{EDA}
\angle{EBF}+\angle{FBC}=\angle{FDE}+\angle{EDA}
\angle{FBC}=\angle{EDA}・・・④

平行四辺形の対辺は等しい性質より、
DA=BC・・・⑤

①④⑤より1辺とその両端の角がそれぞれ等しいので\triangle{ADE} \equiv\triangle{CBF}

(2)次の図形の\triangle{CDA}と等しい面積の三角形、\triangle{DAE}と等しい面積の三角形を答えなさい。
CDを底辺とする三角形の高さはAB//DCよりABDCの距離になりAB上の点Fをとると\triangle{CDF}は全て高さの等しい三角形となります。
これはA以外ではBのみであるから、\triangle{CDA}と等しい面積の三角形は\triangle{CDB}になります。

\triangle{CDA}\triangle{CDB}の面積は等しいことを示しました。
これらの三角形は共に\triangle{CED}を含みますね。
\triangle{CDA}から\triangle{CED}を除いた三角形の面積と\triangle{CDB}から\triangle{CED}を除いた三角形の面積は等しくなります。
\triangle{DAE}\triangle{CDA}から\triangle{CED}を除いた三角形になります。
\triangle{CDB}から\triangle{CED}を除いた三角形は\triangle{CBE}になります。

よって\triangle{DAE}と等しい面積の三角形は\triangle{CBE}になります。

(3)次の図形でBAの延長線上に点Eを取り、\triangle{EBC}の面積が四角形ABCDの面積と等しくなるようにしたい。
どのように点Eを決めたらよいか答えなさい。
面積を等しくなるように変形するときは平行線の等積変形を使いましょう

どうやって平行線を引くかが問題ですね。
ABの延長線上にEを作ると、必ずDが「切り取られ」ますね。
DをBAの延長線上に移すことを考えます

ABCDの中から2点を選んだ組み合わせで直線が引けます。
この中でDから平行な直線を引いて面積を等積変形できそうなのはCAだけですね。
もうDからCAに平行な直線を引くしかないですね。

平行と面積_3

こうして得られた点Eは\triangle{CAD}の面積を等積変形し、\triangle{CAE}と面積が等しくなります。

平行と面積_4

四角形ABCDの面積は\triangle{BAC}+\triangle{CAD}です。
\triangle{CAE}の面積は\triangle{BAC}+\triangle{CAE}です。
\triangle{CAD}の面積は\triangle{CAE}の面積と等しいので、四角形ABCDの面積と\triangle{CAE}の面積は等しくなります。

よって答えは、「DからCAに平行な直線を引いた直線とBAの延長線上の交点をEとする」です。

終わりに

平行という性質は非常に多くの性質をもっています。
これは平行線を見つけると非常に多くの性質が得られるという事です。
平行線は良い奴です。
平行そうな線があったら平行かどうか確認しましょう。

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