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座標と面積に関するTips | 夢を叶える塾
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座標と面積に関するTips

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座標と面積に関して知っておくと便利な事をいくつかご紹介いたします。

面積の求め方

基本的には複雑な図形の面積は

  1. 面積のわかる形に切ってそれぞれ求めて足し合わせる
  2. 面積のわかる形で覆って余分な面積を引く

という方法でわかる形を作ります。
わかる形を増やしておけると解ける問題がぐっと広がりますね。

Tips

本記事の本題です。

三角形が頂点と他の二点で作られ、それぞれの座標が分かっている場合

それぞれの座標をO(0,0),A(x_a,y_a),B(x_b,y_b)とします。
このとき\triangle{OAB}の面積S
\displaystyle S=\frac{1}{2}|x_a y_b-x_b y_a|
となります。
なお、証明は色々あります。
図形と方程式で直線と点の距離を使える高校二年生であればできます。
一次関数を使った証明は中学二年生であればできます。
図形の面積を求めることのできる小学五年生(ただ、文字を使いますが)であればできます。
是非挑戦してみてください。

三角形のそれぞれの座標が分かっている場合

それぞれの座標をA(x_a,y_a),B(x_b,y_b),C(x_c,y_c)とします。
この三角形を頂点に平行移動すれば先の公式が使えますね。
(-x_a,-y_a)平行移動すると、A(0,0),B(x_b-x_a,y_b-y_a),C(x_c-x_a,y_c-y_a)となります。
\triangle{OAB}の面積S
\displaystyle S=\frac{1}{2}|(x_b-x_a)(y_c-y_a)-(x_c-x_a)(y_b-y_a)|
となります。
ちなみにこのような公式は覚えずに、上記の「頂点に平行移動して使おう」という発想が重要であり、それだけ覚えておいた方が良いでしょう。

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