こんにちは。相城です。さて、2020年の2月に行われなした、神奈川県の入試問題より、平行四辺形と面積の問題です。それではどうぞ。
下の図のような平行四辺形ABCDがあり、辺BC上に点Eを辺BCと線分AEが垂直に交わるようにとり、辺AD上に点FをABAFとなるようにとる。
また、線分BFと線分AEとの交点をG、線分BFと線分ACとの交点をHとする。
AB15cm、AD
25cm、
のとき、三角形AGHの面積を求めなさい。
![](https://mathtext.info/blog/wp-content/uploads/2020/02/1yohaku.png)
答え
![Rendered by QuickLaTeX.com \dfrac{45}{2}](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-6e1a9c7b692e3bc20860386867d3c507_l3.png)
![Rendered by QuickLaTeX.com ^2](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-f5665a381aebd5b9ce97a73c9f8da8cd_l3.png)
三平方の定理よりAC
![Rendered by QuickLaTeX.com =](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-168cbc7066049ab4eed81c42c40faad5_l3.png)
△ABCの面積の関係より
![Rendered by QuickLaTeX.com 15\times20=25\times](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-4591e5fbe0af5b1c75126a76b125d7c3_l3.png)
AE
![Rendered by QuickLaTeX.com =](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-168cbc7066049ab4eed81c42c40faad5_l3.png)
![Rendered by QuickLaTeX.com =](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-168cbc7066049ab4eed81c42c40faad5_l3.png)
また△AGF∽△EGBでAG : EG
![Rendered by QuickLaTeX.com =](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-168cbc7066049ab4eed81c42c40faad5_l3.png)
![Rendered by QuickLaTeX.com =](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-168cbc7066049ab4eed81c42c40faad5_l3.png)
△AHF∽△CHBよりAH : CH
![Rendered by QuickLaTeX.com =](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-168cbc7066049ab4eed81c42c40faad5_l3.png)
![Rendered by QuickLaTeX.com =](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-168cbc7066049ab4eed81c42c40faad5_l3.png)
①、②より△AGHは△AECの
![Rendered by QuickLaTeX.com \dfrac{5\times3}{8\times8}=\dfrac{15}{64}](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-16c368a8cf9a2b459df2339051054bf1_l3.png)
![Rendered by QuickLaTeX.com =25-9=16](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-b7169cb9a1d848d05a3a6ae626668736_l3.png)
よって、求める面積は
△AGH
![Rendered by QuickLaTeX.com =16\times12\times\dfrac{1}{2}\times\dfrac{15}{64}=\dfrac{45}{2}](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-3533ce055f9c14baf3358b8c057117b1_l3.png)
![Rendered by QuickLaTeX.com \dfrac{45}{2}](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-6e1a9c7b692e3bc20860386867d3c507_l3.png)
![Rendered by QuickLaTeX.com ^2](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-f5665a381aebd5b9ce97a73c9f8da8cd_l3.png)