こんにちは。面積比の問題です。
【問題】図のように, ,
,
のとき, 辺BAの延長上に
となるように点Gをとる。また, GEとADの交点をHとする。
このとき, 台形ABEHの面積は, 平行四辺形ABCDの面積の何倍になるか, 求めなさい。
【和歌山県一部抜粋】
【解答】倍
【解説】
高さが等しい台形と平行四辺形の面積比は,
台形の(上底+下底):(平行四辺形の底辺)
![Rendered by QuickLaTeX.com \times2\cdots\maru1](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-102642794ad3f231491448c62938e760_l3.png)
で求められる。平行四辺形の底辺は
![Rendered by QuickLaTeX.com 5\, \text{cm}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-3954b3dad1597eb553837f0f38cc2383_l3.png)
![Rendered by QuickLaTeX.com 3\,\text{cm}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5dcc9cd584ee58117fc0cd75651002a1_l3.png)
![Rendered by QuickLaTeX.com \text{AH}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-3b59fe5d106ecb0ddbc998f7531900c7_l3.png)
![Rendered by QuickLaTeX.com \bigtriangleup{\text{GAH}}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-e57098cd4f25b59c6c9d9305afe18d60_l3.png)
![Rendered by QuickLaTeX.com \bigtriangleup{\text{GBE}}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5d2d4ff4e20c5e28064bae5a136db6a5_l3.png)
![Rendered by QuickLaTeX.com \text{GA} : \text{GB} = 1 : 3](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-846f5566a41b5f7e5681a20d9353df9e_l3.png)
![Rendered by QuickLaTeX.com 1 : 3 = \text{AH} : 3](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-7895d1208ade700a8fe6d9472021a5d4_l3.png)
![Rendered by QuickLaTeX.com \text{AH}=1\, \text{cm}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-8009310723fdbec25e6c02166ed36b4e_l3.png)
よって, 台形ABEHの上底と下底の和は
![Rendered by QuickLaTeX.com 1+3=4](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-6f09b0d74130e0852833a97444b3cdc8_l3.png)
![Rendered by QuickLaTeX.com 5\times2=10](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-7cb9b473ccbeebbe98eee9abd7b475b3_l3.png)
ゆえに求める答えは,
![Rendered by QuickLaTeX.com 4\div10=\dfrac25](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-94916e9c30a08f6d4763cf3aca49c0cd_l3.png)
![Rendered by QuickLaTeX.com \dfrac25](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-3b58ac65381957a33c12ee1c51abacac_l3.png)
![Rendered by QuickLaTeX.com \cdots](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-f92a08ced98be124fef39e8b49d7144a_l3.png)
※【補足】
![Rendered by QuickLaTeX.com \maru1](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-c8b8ec9c0d15342374d474f3407d687d_l3.png)