こんにちは。今回の基礎学ですが, 難しかったですか?難しいと見えた方は練習量が足りていないんでしょうね。ここのサイトで繰り返し練習を積んでみてください。今回は平面図形の問題を解説してみたいと思います。
下の図の, は
の二等辺三角形である。点Aを通り辺BCに平行な直線をひき, その直線上に
となる点D,
となる点Eをとる。
点Cと点D, Eをそれぞれ結び, CDとAB, CEとABとの交点をそれぞれF, Gとする。
次の(1)~(3)に答えなさい。
(1)
![Rendered by QuickLaTeX.com \bigtriangleup{\text{ADF}}\equiv\bigtriangleup{\text{BCF}}](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-2063b33b8f70e578d95de4e981750956_l3.png)
(2)
![Rendered by QuickLaTeX.com \kaku{DAF}=a\Deg](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-e8864ff2b4e22cd8f89f0752be977f73_l3.png)
![Rendered by QuickLaTeX.com \kaku{DCE}](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-2e6def74d58032d3f4d7e957b08931b2_l3.png)
![Rendered by QuickLaTeX.com a](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-6e4239cd9fe5a53bc98c863c75818b12_l3.png)
(3) 点Eと点Bを結ぶ。
![Rendered by QuickLaTeX.com \text{AD} : \text{DE} = 2 : 1](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-8030fd3ed44ff484279af4594038d338_l3.png)
![Rendered by QuickLaTeX.com \text{FG} : \text{GB} = 1 : 4](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-54f2abb61f9a91f47534cb0bdfb3666f_l3.png)
![Rendered by QuickLaTeX.com \bigtriangleup{\text{CFG}}](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-fd041c7a0e6551d2708d9d62666086d7_l3.png)
【2023年徳島県中3第2回基礎学力テスト】
(1) と
で
仮定より, より, 錯角は等しいので,
より,
1組の辺とその両端の角がそれぞれ等しいので,
(2)
![Rendered by QuickLaTeX.com \kaku{DAF}=\kaku{CBF}=a](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-a6d520c819d61195b9b4e2e436c5d50f_l3.png)
![Rendered by QuickLaTeX.com \kaku{CBF}=\kaku{CAB}=a](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-d888baddc01c4aa42cc007c9cd1bbacc_l3.png)
![Rendered by QuickLaTeX.com \text{CA=CB}](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-892b9c3587dbe85b90f364118baf7f50_l3.png)
これより,
![Rendered by QuickLaTeX.com \kaku{CAD}=2a=\kaku{CED}](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-23a9d101dd857b7b8f31d1d3824104b5_l3.png)
![Rendered by QuickLaTeX.com \text{CA=CE}](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-a3ecda23bf1b49625fa62d1dde503b47_l3.png)
![Rendered by QuickLaTeX.com \text{EA//BC}](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-f29c57187a98160821406ee25d3f4cd3_l3.png)
![Rendered by QuickLaTeX.com \kaku{CED}=\kaku{ECB}=2a\cdots\maru1](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-19fb8171ae8eb6953e7687cb7a8a16fb_l3.png)
ここで
![Rendered by QuickLaTeX.com \bigtriahgleup{\text{ACD}}](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-d0d57659bb28b206d10639221d5e8644_l3.png)
![Rendered by QuickLaTeX.com \kaku{AFC}=90\Deg](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-44525613f64c01a4e9351433d3eadbdf_l3.png)
![Rendered by QuickLaTeX.com \bigtriangleup{\text{AFC}}](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-b63dcedab3a190fbb9b9bb5897b7d616_l3.png)
![Rendered by QuickLaTeX.com \kaku{CAF}=a](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-affdc2e5c917bd38b9b5808d7200bf98_l3.png)
![Rendered by QuickLaTeX.com \kaku{ACF}=90-a](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-e619b6df0ccae8c8099c172a65f88331_l3.png)
また,
![Rendered by QuickLaTeX.com \bigtriangleup{\text{ABC}}](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-a243832fb54297cd6618de16df59534b_l3.png)
![Rendered by QuickLaTeX.com \kaku{ACB}](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-5588edce4848f2ce40220aba879f8791_l3.png)
![Rendered by QuickLaTeX.com \kaku{AFC}=90\Deg](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-44525613f64c01a4e9351433d3eadbdf_l3.png)
直線CFは頂角の二等分線である。
したがって,
![Rendered by QuickLaTeX.com \kaku{ACF}=\kaku{BCF}=90-a\cdots\maru2](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-c0ccd0abc02f8af4bc948de22d169399_l3.png)
![Rendered by QuickLaTeX.com \maru1, \maru2](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-17f7275e5547cb30256f14fe6f9231f7_l3.png)
![Rendered by QuickLaTeX.com \kaku{DCE}=\kaku{BCF}-\kaku{ECB}=90-a-2a=90-3a](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-efd3344f7266b193376347f7dc6b0f03_l3.png)
![Rendered by QuickLaTeX.com (90-3a)\Deg\cdots](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-c55c2949b31d224ebf11b924ef051802_l3.png)
(3)
EとB, DとBを結ぶ。
(2)から四角形ACBDはひし形である。したがって, 対角線で区切られた4つの三角形の面積はすべて等しい。
また,
![Rendered by QuickLaTeX.com \text{AD : DE = 2 : 1}](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-8ba47bd881abe4025cafa409e73a5734_l3.png)
![Rendered by QuickLaTeX.com \text{FG : GB = 1 : 4}](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-93dfb89cae75ee6ce82387afca9fe2ae_l3.png)
![Rendered by QuickLaTeX.com \text{DE : BC = 1 : 2}](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-8c9e08a90d56b4d41beb9088acf6234f_l3.png)
![Rendered by QuickLaTeX.com \text{GB : FG : FA = 4 : 1 : 5}](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-77fab7ce858344dd8319e549f56ad2c3_l3.png)
![Rendered by QuickLaTeX.com \bigtriangleup{\text{BCG}} : \bigtriangleup{\text{CFG}} : \bigtriangleup{\text{ACF}} = 4 : 1 : 5](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-2f82fefe1b2f77b97545420ba8b1b42e_l3.png)
したがって,
![Rendered by QuickLaTeX.com \bigtriangleup{\text{ADF}}=\bigtriangleup{\text{DBF}}=\bigtriangleup{\text{ACF}}=5](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-cfb4350fd729fd030bae7e8f9f759a50_l3.png)
四角形EBCDは台形で上底と下底の比は
![Rendered by QuickLaTeX.com \text{DE : BC = 1 : 2}](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-8c9e08a90d56b4d41beb9088acf6234f_l3.png)
![Rendered by QuickLaTeX.com \bigtriangleup{\text{BCD}}=10](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-9ab3a8c943681256ecca7ef6ac24badb_l3.png)
![Rendered by QuickLaTeX.com \bigtriangleup{\text{BDE}}=5](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-a086dc67d19debfc7ed35168d93fdef8_l3.png)
以上より, 四角形BCAEは25,
![Rendered by QuickLaTeX.com \bigtriangleup{\text{CFG}}=1](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-fbab6a672b333e6621ad858d18c34ce5_l3.png)
![Rendered by QuickLaTeX.com 25\div1=25](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-cfcfd3faa5b5e2403c259a1dc2ee9cb1_l3.png)
25倍
![Rendered by QuickLaTeX.com \cdots](https://mathtext.info/blog/wp-content/ql-cache/quicklatex.com-f92a08ced98be124fef39e8b49d7144a_l3.png)