こんにちは。今回は2022年に行われた徳島県の公立高校の入試問題から大問5の平面図形をやってみようと思います。
【問題】図1, 図2のように, AB4cm,
である長方形ABCDを, ある線分を折り目として折り返したものがある。(1)・(2)に答えなさい。
(1) 図1のように, 長方形ABCDを, 辺CD上の点Eと頂点Bを結んだ線分BEを折り目として, 頂点Cが辺AD上にくるように折り返したとき, 頂点Cが移る点をFとする。(a)・(b)に答えなさい。
図1
(a)
![Rendered by QuickLaTeX.com \kaku{ABF}=50^{\circ}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-7057a91e86f7dc86f9c55444c12d28c8_l3.png)
![Rendered by QuickLaTeX.com \kaku{BEF}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-4cdf9029a6a1a9f51ff685bf370b0b09_l3.png)
(b) DE : EC
![Rendered by QuickLaTeX.com = 7 : 9](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-8590449328753af5a200a90009f8932b_l3.png)
(2) 図2のように, 長方形ABCDを, 対角線BDを折り目として折り返したとき, 頂点Cが移る点をP, 辺ADと線分BPとの交点をQとする。(a)・(b)に答えなさい。
図2
(a) △ABQ
![Rendered by QuickLaTeX.com \equiv](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-366b6092f94455eb2f8df6d17fbeaf46_l3.png)
(b) 対角線BDの中点をR, 線分ARと線分BPとの交点をSとする。AD
![Rendered by QuickLaTeX.com =](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-168cbc7066049ab4eed81c42c40faad5_l3.png)
「2022年徳島県公立高校入試」
(1)(a)
図より
![Rendered by QuickLaTeX.com 70^{\circ}\cdots](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-49501a85f2256481935b29b0c92edd54_l3.png)
(1)(b)
DE : EC
![Rendered by QuickLaTeX.com = 7 : 9](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-8590449328753af5a200a90009f8932b_l3.png)
![Rendered by QuickLaTeX.com =7x](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-f471cb77dbeebfe2ab68d5f56093945e_l3.png)
![Rendered by QuickLaTeX.com =9x](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-ce8d6d7eb4a97c52047d7a2d4762256b_l3.png)
このとき,
![Rendered by QuickLaTeX.com 7x+9x=4](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-94bc6d12ce74f2ccf5a88446b4fbdce9_l3.png)
![Rendered by QuickLaTeX.com x=\dfrac14](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-cc843a6536c70f22eb95269ea35b7f2f_l3.png)
したがって, DE
![Rendered by QuickLaTeX.com =\dfrac74](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-a42625e8189b65d49232b7e5b4dd73e5_l3.png)
![Rendered by QuickLaTeX.com =\dfrac94](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-4702e3622edc29074d5219727f50def6_l3.png)
![Rendered by QuickLaTeX.com =](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-168cbc7066049ab4eed81c42c40faad5_l3.png)
よって, EF
![Rendered by QuickLaTeX.com =\dfrac94](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-4702e3622edc29074d5219727f50def6_l3.png)
![Rendered by QuickLaTeX.com \cdots](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-f92a08ced98be124fef39e8b49d7144a_l3.png)
(2)(a)
△ABQと△PDQで,
仮定より,
AB
![Rendered by QuickLaTeX.com =](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-168cbc7066049ab4eed81c42c40faad5_l3.png)
![Rendered by QuickLaTeX.com \cdots\maru1](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-10f8e05c8728d08fd87c21a42bb15290_l3.png)
![Rendered by QuickLaTeX.com \kaku{A}=\kaku{P}=90^{\circ}\cdots\maru2](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-66b52bf7ab2d6643632435f8ef623f48_l3.png)
対頂角は等しいので,
![Rendered by QuickLaTeX.com \kaku{AQB}=\kaku{PQD}\cdots\maru3](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-c656aac27d3fe14bf9bd19cab357c743_l3.png)
![Rendered by QuickLaTeX.com \maru2, \maru3](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-d9f7bb833a4c52d61caad744d8d1c1b9_l3.png)
![Rendered by QuickLaTeX.com \kaku{QBA}=\kaku{QDP}\cdots\maru4](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-0e7a4b71e9fca3e74da76dea4770fc4c_l3.png)
![Rendered by QuickLaTeX.com \maru1, \maru2, \maru4](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-ea658c69ecb1a6ff137b3d6f26519fd7_l3.png)
1組の辺とその両端の角がそれぞれ等しいので,
△ABQ
![Rendered by QuickLaTeX.com \equiv](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-366b6092f94455eb2f8df6d17fbeaf46_l3.png)
【別解】手間はかかりますが, △QBDが二等辺三角形を示し, QB
![Rendered by QuickLaTeX.com =](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-168cbc7066049ab4eed81c42c40faad5_l3.png)
(2)(b)
まず最初に線分ARはCの方に延長して, 線分ACとしておきます。理由は△AQS∽△CBSをつくるためです。
次に,
![Rendered by QuickLaTeX.com \text{AQ}=\text{PQ}=x](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-f98705c014e38cfaf4c3b80f3dde3def_l3.png)
![Rendered by QuickLaTeX.com \text{QD}=\text{QB}=12-x](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-894e9c9586b64c66f5bfad7e833f5fbe_l3.png)
![Rendered by QuickLaTeX.com x](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5f16b0dcec027c9742e11d99170299a8_l3.png)
![Rendered by QuickLaTeX.com (12-x)^2=x^2+16](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-7ecd9b1c9909e294c8b0d284aa8e8f1f_l3.png)
![Rendered by QuickLaTeX.com 24x=128](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-a3b3119c9d83e0a2f9eac31c1a8b623e_l3.png)
![Rendered by QuickLaTeX.com x=\dfrac{16}{3}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-b9df0f7886191f068cec504c0ed816c6_l3.png)
![Rendered by QuickLaTeX.com \text{AQ}=\text{PQ}=\dfrac{16}{3}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-b819e062f78fe21478c0f47dcd62bacc_l3.png)
![Rendered by QuickLaTeX.com \text{QD}=\text{QB}=12-\dfrac{16}{3}=\dfrac{20}{3}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-922ce98b6a86b13f652b981a979a10f0_l3.png)
△AQS∽△CBSで相似比は, AQ : CB
![Rendered by QuickLaTeX.com =\dfrac{16}{3} : 12= 4: 9](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-6d72555d280366ea89c86a5bb87f6619_l3.png)
したがって, QS : BS
![Rendered by QuickLaTeX.com = 4 : 9](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-c0d2998f362ed4e0f4235557b7034028_l3.png)
よって,
QS
![Rendered by QuickLaTeX.com =\dfrac{20}{3}\times\dfrac{4}{13}=\dfrac{80}{39}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-afffdf8750e4b1e0c527f77d02069200_l3.png)
BS
![Rendered by QuickLaTeX.com =\dfrac{20}{3}\times\dfrac{9}{13}=\dfrac{60}{13}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-302e0cef67f2ebab24343e0842b38619_l3.png)
これから, SP
![Rendered by QuickLaTeX.com =\dfrac{16}{3}+\dfrac{80}{39}=\dfrac{288}{39}=\dfrac{96}{13}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-8533a0c9d68340b21adb38b9f46b6ede_l3.png)
したがって, BS : SP
![Rendered by QuickLaTeX.com = \dfrac{60}{13} : \dfrac{96}{13}= 5 : 8](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-9e2fc3341ccbdac9a5c44e3e4d551824_l3.png)
これから,
△BRS : △BDP
![Rendered by QuickLaTeX.com = 1\times5 : (1+1)\times(5+8)= 5 : 26](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-31503da18706a2b8c27c0475fb8eb14a_l3.png)
したがって,
△BRS : 四角形RDPS
![Rendered by QuickLaTeX.com = 5 : (26-5) = 5 : 21](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-158f87987b8204bd3d3db95e73712d83_l3.png)
よって,
![Rendered by QuickLaTeX.com 21\div5=\dfrac{21}{5}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-7a610bf509b1e4f2c432478553e56f0a_l3.png)
![Rendered by QuickLaTeX.com \dfrac{21}{5}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-77894195b18614000de8cd36bb3ae93d_l3.png)
![Rendered by QuickLaTeX.com \cdots](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-f92a08ced98be124fef39e8b49d7144a_l3.png)
【別解1】△BPD
![Rendered by QuickLaTeX.com \equiv](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-366b6092f94455eb2f8df6d17fbeaf46_l3.png)
![Rendered by QuickLaTeX.com =](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-168cbc7066049ab4eed81c42c40faad5_l3.png)
![Rendered by QuickLaTeX.com =](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-168cbc7066049ab4eed81c42c40faad5_l3.png)
![Rendered by QuickLaTeX.com =](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-168cbc7066049ab4eed81c42c40faad5_l3.png)
【別解2】△BPD
![Rendered by QuickLaTeX.com \equiv](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-366b6092f94455eb2f8df6d17fbeaf46_l3.png)
![Rendered by QuickLaTeX.com \kaku{CBR}=\kaku{SBR}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-70873cd61acced79b5ebeecc0581297f_l3.png)
【余談】余力のある方はせっかくなんで, (1)でADの長さを求めてみてもいいでしょう。答えは
![Rendered by QuickLaTeX.com \dfrac92\sqrt{2}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-f0243c6a762a3c76ff712ccfdd98815e_l3.png)