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【平面上の内分点,外分点の座標】
2点 (1) m : nに内分する点Pの座標は 特に,中点(1:1に内分する点)Mの座標は (2) m : n (m≠n)に外分する点Qの座標は ![]()
以下の解説は,多くの教科書に書かれているものとほぼ同じです.
(1)←
以下においては,
中学2年生で習う平行線の性質,または中学校3年生で習う相似図形の性質を使うと,右図においてAP : BP=m : nのとき,A'P' : B'P'=m : n したがって y座標についても同様に示すことができる. 中点の公式はm=n=1とすると得られる. ![]()
以下においては,
中学2年生で習う平行線の性質,または中学校3年生で習う相似図形の性質を使うと,右図においてAQ : BQ=m : nのとき,A'Q' : B'Q'=m : n したがって y座標についても同様に示すことができる. ⇒このように,内分公式のnのところに−nを代入すると,外分公式になる |
※よくある間違いに注意
座標の値が
(1) x座標とy座標を「混ぜない」こと
(2) mとnは「クロスして掛ける」こと (3) 外分公式では,「nだけを負にする」こと (1) x座標とy座標を「混ぜない」こと
#1つの点のx座標とy座標を混ぜてしまう間違いが多い# 〇:正しい計算↓ ![]() ![]() ![]()
【例1】
間違い ⇒ 2点A(1, 2), B(3, 4)を結ぶ線分ABを5 : 6に内分する点の座標は 正しい ⇒ (2) mとnは「クロスして掛ける」こと
![]() であるから,分母にあるm,nを分子に掛けるときは「クロスして掛ける」ことになります. (見かけにダマされてはいけない「意地悪」「へそ曲げ」の公式となっています) 証明は,上で行ったので,ここでは別の解釈を示してみます. のように分けて見ると,2つの係数の和は1です. このとき,
例えば,
になっています.このとき, ⇒大きい数字が掛けてある方の点に近くなる (3) 外分公式では,「nだけを負にする」こと
外分公式のx座標,y座標のいずれも分母と分子の両方に−1を掛けても,分数には1を掛けることになって,元の式と同じ値になるから と書いてもよい.さらに,分母が負の数になってしまうと,計算間違いしやすくなるので,これを防ぐために「mとnの符号は大きい方を正に,小さい方を負にする」…(*3)と教える先生もいます. (*1)(*2)(*3)とも正しく,実際,筆者も「どれでもよい」と教えてきましたが,教育心理的にはそれは「できる生徒向けの説明」かもしれません.同値な式を一巡聞いておくのはよいことですが,まとめとしては1つの公式を確実に身に着ける方がよいでしょう.苦手な生徒向けとしては,正しいからと言って,何を言ってもよいとは限らず,揺れのある表現で言われると,判断で迷って,形が混ざってしまうミスを誘発しやすいので, 外分公式では,「nだけを負にする」 と決める方が間違いが少なくなると考えられます. 特に,(*1)(*2)を並べて教えると,「外分では,m,nを負にすればいいんだな」と単純化して覚える生徒が見られ, にしてしまうことがあります.実際には,(*4)の分母分子に−1を掛けると分かるように,次の内分公式と同じものになります.[(*1)(*2)(*3)は外分公式,(*4)は内分公式] ⇒2つとも負にすると,元に戻ってしまう. |
公式を確実に身に着けるための問題
【問題1】 次の各点の座標を求めてください.(選択肢の中から正しいものをクリック)
※暗算でやるのは無理ですから,別途計算用紙で計算してから答えてください.まぐれ当たりで合っても実力はつきません.
(1)
2点A(3, 4),B(5, 6)を結ぶ線分ABを2:1に内分する点の座標 |
(2)
2点A(3, 5),B(4, 6)を結ぶ線分ABを1:2に内分する点の座標 |
(3)
2点A(3, −5),B(4, −6)を結ぶ線分ABを3:2に内分する点の座標 |
(4)
2点A(−3, 5),B(−4, 6)を結ぶ線分ABを1:2に外分する点の座標 |
(5)
2点A(3, −4), B(5, −6)を結ぶ線分ABを2:3に外分する点の座標 |
(6)
2点A(−3, −4), B(−5, 6)を結ぶ線分ABを1:3に外分する点の座標 |
(7)
2点A(−3, −5), B(5, 6)を結ぶ線分ABを2:5に内分する点の座標 |
(8)
2点A(3, 4), B(−5, −6)を結ぶ線分ABを3:2に外分する点の座標 |
ゆっくり考える問題(教科書レベル)
【問題2】 (選択肢の中から正しいものをクリック)
(1)
点A(−1, 2)に関して点B(3, −4)と対称な点Pの座標を求めてください. ![]() Aに関してBと対称な点がPであるとは,右図の茶色で示したように,点Aを対称の中心としてBとPが点対称であるということ. このためには,BPの中点がAになればよい. P(x, y)とおくと を解くと [未知数を使わずに,算数で解く方法]
Aに関してBと対称な点がPであるとは,右図の灰色で示したように,BAを2:1に外分する点がPであるということ |
[平行四辺形の2つの対角線は,互いに他を二等分することを使って,次の問題を解いてください]
(2) 4点A, B, C, Dをこの順にたどると平行四辺形になるという. A(2, 2), B(−1, 1), C(0, 3)のとき,点Dの座標を求めてください.
ACの中点とBDの中点が一致すればよい
より ![]() |
(3) 少し難しいかも
△ABCの頂点A, B, Cについて,ABの中点をP,PCの中点をQ,BCを2 : 1に内分する点をRとすると,QはARを何対何に内分しますか. |
(4) 少し難しいかも
m>n>0とするとき,線分ABをm : nに内分する点をP,m : nに外分する点をQとおくと,点Bは線分PQを何対何に内分しますか. |
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