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円と接線の問題の解法 | 夢を叶える塾
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円と接線の問題の解法

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円の2本の接線が作る線分の長さの性質を使った問題の解法です。

基本問題

(1)次の図のQCの長さを求めなさい。
円と接線_1
AR=2,BC=5,RB=3

(2)(1)の図でAB=15,BC=17,CA=8で、\triangle{ABC}の面積が60のとき、内接円の半径を求めなさい。

解き方

(1)は円の2本の接線が作る線分の長さは等しくなります。
(2)は内接円の半径を使って三角形の面積を表します。

解説

(1)次の図のQCの長さを求めなさい。
このコーン型の円の2本の接線が作る線分の長さは等しくなります
AR=AQ,QC=CP,PB=BRになります。

RB=3なのでBP=3になります。
BC=CP+PBなので5=CP+3CP=2になります。
QC=CPなのでQC=2になります。

(2)(1)の図でAB=15,BC=17,CA=8で、\triangle{ABC}の面積が60のとき、内接円の半径を求めなさい。

面積を内接円の半径rを使って表します。
そのために、半径がわかるように図に線を描いておいたほうが良いですね。

円と接線_2

この状態ですと面積の式は四角形の面積の和ですが、これら四角形の面積は求める事が難しそうです。
更に面積を細かく分けて求めます。

四角形が求めにくいので三角形にしましょう。
対角線で切り取れば四角形になりますが、RP,PQ,QRで切っても面積を求めることは難しそうです。
AO,BO,COはどうでしょうか?

円と接線_3

\triangle{ABC}=\triangle{ARO}+\triangle{AQO}+\triangle{CQO}+\triangle{CPO}+\triangle{BPO}+\triangle{BRO}
半径rを使った面積の式になりそうですね。

AB,BC,CAの長さはわかっています。
ABを底辺とする高さROの面積は半径rを使って表すことができます。
同じようにBC,CAを底辺三角形の面積も半径rを使って表すことができます。
くっつけてしまいましょう。

\triangle{ABC}=\triangle{ARO}+\triangle{BRO}+\triangle{AQO}+\triangle{CQO}+\triangle{CPO}+\triangle{BPO}
=\triangle{ABO}+\triangle{CAO}+\triangle{BCO}
=(AB\times r \div 2)+(CA\times r \div 2)+(BC\times r \div 2)
=(AB+BC+CA)\times r \div 2

これは三角形の各辺と内接円の半径から面積を求める式になっています
\triangle{ABC}の面積が60、AB=15,BC=17,CA=8を代入します。

60=(15+17+8)\times r \div 2
120=40\times r
r=3

よって内接円の半径は3になります。

終わりに

高校入試の問題では図形で円の性質が良く出題されています。
「円に2本の接線が引かれている」ことが読み取れたら、基本問題の(1)の性質を使うんだなと思ってください。
(2)の内接円の半径の問題も公式を知っていると早く求める事ができます。
覚えることもなく、スラッと導いて公式から求められると良いですね。

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