こんにちは。相城です。さて2020年度3月10日に行われた徳島県の高校入試の問題からです。それではどうぞ。
下の図のように、半径が15cmの円Oの周上に4点A、B、C、Dがあり、AC=ADである。また、弦ACはの二等分線であり、弦ACと弦BDの交点をEとする。(1)~(3)に答えなさい。ただし、円周率は
とします。
(1) のとき、(a)、(b)に答えなさい。
(a) の大きさを求めなさい。
(b) 点Aを含まないおうぎ形OBCの面積を求めなさい。
(2) △ABC△AEDを証明しなさい。
(3) 点Cを含まないの長さが
cmのとき、点Bを含まない弧ADの長さを求めなさい。
![](https://www.mathtext.info/blog/wordpress/wp-content/uploads/2020/02/1yohaku.png)
答え
(1)
(a)![Rendered by QuickLaTeX.com 70^{\circ}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-38feb769e5ba7795cb3f68043b474c97_l3.png)
とする。CDを結ぶと△ACDが頂角
、底角
の二等辺三角形になり、
に対する円周角より、![Rendered by QuickLaTeX.com \angle{\text{ACD}}=\angle{\text{ABD}}=70^{\circ}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-e89e93c5fc344d2d71443bc5d1ac88fc_l3.png)
(b)
cm![Rendered by QuickLaTeX.com ^2](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-f5665a381aebd5b9ce97a73c9f8da8cd_l3.png)
![Rendered by QuickLaTeX.com \angle{\text{BOC}}=2\angle{\text{BAC}}=80^{\circ}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-4bb96b6a3cef637b5aab54c05adc5dd5_l3.png)
よって、![Rendered by QuickLaTeX.com 15\times15\times\pi\times\dfrac{80^{\circ}}{360^{\circ}}=50\pi](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-c94fd97e476fdff36617a80fc8a0d8e6_l3.png)
cm![Rendered by QuickLaTeX.com ^2](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-f5665a381aebd5b9ce97a73c9f8da8cd_l3.png)
(2)
△ABCと△AEDで、
仮定より
・・・①
AC
AD・・・②
に対する円周角は等しいので、
・・・③
①、②、③より1組の辺とその両端の角がそれぞれ等しいので、
△ABC
△AED
(3)
cm
AとO、BとO、DとOを結ぶ。
中心角
は
![Rendered by QuickLaTeX.com 360^{\circ}\times\dfrac{8\pi}{30\pi}=96^{\circ}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-2eb2c7fc59b57ac930199963a80e200d_l3.png)
このとき、![Rendered by QuickLaTeX.com \angle{\text{OAB}}=\angle{\text{OBA}}=42^{\circ}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-6ae8aa2cee2f02939e5440b7a729e8b4_l3.png)
△OAC
△OADとなり
とおくと、
より、
。このとき、
であるから、
。よって、![Rendered by QuickLaTeX.com \angle{\text{AOD}}=180^{\circ}-14^{\circ}\times2=152^{\circ}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-289932b3965b767a15a7ef012c036605_l3.png)
ゆえに、![](https://www.mathtext.info/blog/wordpress/wp-content/uploads/2020/12/koad4.png)
![Rendered by QuickLaTeX.com =30\pi\times\dfrac{152^{\circ}}{360^{\circ}}=\dfrac{38}{3}\pi](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-fc4cd9aaed492271c8393766064f19fb_l3.png)
cm
(a)
![Rendered by QuickLaTeX.com 70^{\circ}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-38feb769e5ba7795cb3f68043b474c97_l3.png)
![Rendered by QuickLaTeX.com \angle{\text{BAD}}=80^{\circ}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-03eabb3844c03a0fc255d6f2e408774a_l3.png)
![Rendered by QuickLaTeX.com 40^{\circ}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-94fd1d3a5461227610dcd87250f7b49d_l3.png)
![Rendered by QuickLaTeX.com 70^{\circ}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-38feb769e5ba7795cb3f68043b474c97_l3.png)
![](https://www.mathtext.info/blog/wordpress/wp-content/uploads/2020/12/koad4.png)
![Rendered by QuickLaTeX.com \angle{\text{ACD}}=\angle{\text{ABD}}=70^{\circ}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-e89e93c5fc344d2d71443bc5d1ac88fc_l3.png)
(b)
![Rendered by QuickLaTeX.com 50\pi](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-cb8b460fac2c3a45f11a79e8c25bfbc9_l3.png)
![Rendered by QuickLaTeX.com ^2](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-f5665a381aebd5b9ce97a73c9f8da8cd_l3.png)
![Rendered by QuickLaTeX.com \angle{\text{BOC}}=2\angle{\text{BAC}}=80^{\circ}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-4bb96b6a3cef637b5aab54c05adc5dd5_l3.png)
よって、
![Rendered by QuickLaTeX.com 15\times15\times\pi\times\dfrac{80^{\circ}}{360^{\circ}}=50\pi](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-c94fd97e476fdff36617a80fc8a0d8e6_l3.png)
![Rendered by QuickLaTeX.com 50\pi](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-cb8b460fac2c3a45f11a79e8c25bfbc9_l3.png)
![Rendered by QuickLaTeX.com ^2](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-f5665a381aebd5b9ce97a73c9f8da8cd_l3.png)
(2)
△ABCと△AEDで、
仮定より
![Rendered by QuickLaTeX.com \angle{\text{BAC}}=\angle{\text{EAD}}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-94d66caa2a90eae328aeed3539ab7480_l3.png)
AC
![Rendered by QuickLaTeX.com =](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-168cbc7066049ab4eed81c42c40faad5_l3.png)
![](https://www.mathtext.info/blog/wordpress/wp-content/uploads/2020/12/koab4.png)
![Rendered by QuickLaTeX.com \angle{\text{ACB}}=\angle{\text{ADE}}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-d9d583f945d16a723eb18c1c93aea9c6_l3.png)
①、②、③より1組の辺とその両端の角がそれぞれ等しいので、
△ABC
![Rendered by QuickLaTeX.com \equiv](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-366b6092f94455eb2f8df6d17fbeaf46_l3.png)
(3)
![Rendered by QuickLaTeX.com \dfrac{38}{3}\pi](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5ea71257c250fad36bd1b4b237ea9d6a_l3.png)
AとO、BとO、DとOを結ぶ。
中心角
![Rendered by QuickLaTeX.com \angle{\text{AOB}}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-2b14689c4089d8989bd7452ab1736840_l3.png)
![Rendered by QuickLaTeX.com 360^{\circ}\times\dfrac{8\pi}{30\pi}=96^{\circ}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-2eb2c7fc59b57ac930199963a80e200d_l3.png)
このとき、
![Rendered by QuickLaTeX.com \angle{\text{OAB}}=\angle{\text{OBA}}=42^{\circ}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-6ae8aa2cee2f02939e5440b7a729e8b4_l3.png)
△OAC
![Rendered by QuickLaTeX.com \equiv](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-366b6092f94455eb2f8df6d17fbeaf46_l3.png)
![Rendered by QuickLaTeX.com \angle{\text{OAC}}=\angle{\text{OAD}}=\bullet](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-7766e7e2ed4286f1bc50b9d6f2b0ee26_l3.png)
![Rendered by QuickLaTeX.com \angle{\text{BAC}}=\angle{\text{CAD}}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-19b263b0d2352c80784ea9abcb3ecb18_l3.png)
![Rendered by QuickLaTeX.com \angle{\text{BAC}}=\bullet\bullet](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-df304259de0598995337cb0b9d030e7f_l3.png)
![Rendered by QuickLaTeX.com \angle{\text{OAB}}=\angle{\text{BAC}}+\angle{\text{OAC}}=\bullet\bullet\bullet=42^{\circ}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-b2f49312704effb5fb253b17033ac09f_l3.png)
![Rendered by QuickLaTeX.com \bullet=42^{\circ}\div3=14^{\circ}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-a3c394a397d53ea3cfdca4c441605634_l3.png)
![Rendered by QuickLaTeX.com \angle{\text{AOD}}=180^{\circ}-14^{\circ}\times2=152^{\circ}](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-289932b3965b767a15a7ef012c036605_l3.png)
ゆえに、
![](https://www.mathtext.info/blog/wordpress/wp-content/uploads/2020/12/koad4.png)
![Rendered by QuickLaTeX.com =30\pi\times\dfrac{152^{\circ}}{360^{\circ}}=\dfrac{38}{3}\pi](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-fc4cd9aaed492271c8393766064f19fb_l3.png)
![Rendered by QuickLaTeX.com \dfrac{38}{3}\pi](https://mathtext.info/blog/wordpress/wp-content/ql-cache/quicklatex.com-5ea71257c250fad36bd1b4b237ea9d6a_l3.png)
数樂管理人
この解き方以外でも解くことが可能です。以下のリンクからご覧ください。皆はどの解き方かな?
![](https://mathtext.info/blog/wordpress/wp-content/uploads/2020/03/112020tokushima-renritu-160x92.png)