線分((1+√2)/2, (-1-√2)/2)と線分(1/2-(-1-√2)/2, -1/2-(1+√2)/2)の比

線分\(\left(\frac{1+\sqrt{2}}{2},\ \frac{-1-\sqrt{2}}{2}\right)と線分\left(\frac{1}{2}-\frac{-1-\sqrt{2}}{2},\ -\frac{1}{2}-\frac{1+\sqrt{2}}{2}\right)\)の比

線分\(\left(\frac{1+\sqrt{2}}{2},\ \frac{-1-\sqrt{2}}{2}\right)と線分\left(\frac{1}{2}-\frac{-1-\sqrt{2}}{2},\ -\frac{1}{2}-\frac{1+\sqrt{2}}{2}\right)\)の比を求めるのに必要な道具

  1. 二重根号の外し方
  2. 2点間の距離

二重根号の外し方

$$\sqrt{(p+q)+2\sqrt{pq}}=\sqrt{p}+\sqrt{q}$$

足して\(p+q、掛けてpq\)の組み合わせを見つける。

根号内にマイナス記号がある場合は,\(p, qのうちの大きい方を前にして引き算をしないと\sqrt{p}-\sqrt{q}\)が正の数にならないことに注意。

2点間の距離

$$\overline{PQ}=\sqrt{(a_2-a_1)^2+(b_2-b_1)^2}$$

線分\(\left(\frac{1+\sqrt{2}}{2},\ \frac{-1-\sqrt{2}}{2}\right)\)の距離

\(\sqrt{\left(\frac{1+\sqrt{2}}{2},\ \frac{-1-\sqrt{2}}{2}\right)^2}\)

$$=\sqrt{\frac{1+2+2\sqrt{2}}{4}+\frac{1+2+2\sqrt{2}}{4}}=\sqrt{\frac{6+4\sqrt{2}}{4}}=\frac{\sqrt{6+4\sqrt{2}}}{2}$$

$$=\frac{\sqrt{6+2\sqrt{8}}}{2}=\frac{\sqrt{4}+\sqrt{2}}{2}=\frac{2+\sqrt{2}}{2}$$

線分\(\left(\frac{1}{2}-\frac{-1-\sqrt{2}}{2},\ -\frac{1}{2}-\frac{1+\sqrt{2}}{2}\right)\)の距離

\(\frac{1}{2}-\frac{-1-\sqrt{2}}{2}\)

$$=\frac{1+1+\sqrt{2}}{2}=\frac{2+\sqrt{2}}{2}$$

\(-\frac{1}{2}-\frac{1+\sqrt{2}}{2}\)

$$=\frac{-1-1-\sqrt{2}}{2}=\frac{-2-\sqrt{2}}{2}$$

\(\sqrt{\left(\frac{1}{2}-\frac{-1-\sqrt{2}}{2},\ -\frac{1}{2}-\frac{1+\sqrt{2}}{2}\right)^2}\)

$$=\sqrt{\frac{4+2+4\sqrt{2}}{4}+\frac{4+2+4\sqrt{2}}{4}}=\sqrt{\frac{12+8\sqrt{2}}{4}}=\frac{\sqrt{12+8\sqrt{2}}}{2}$$

$$=\frac{\sqrt{12+2\sqrt{32}}}{2}=\frac{\sqrt{8}+\sqrt{4}}{2}=\frac{2\sqrt{2}+2}{2}=\sqrt{2}+1$$

線分\(\left(\frac{1+\sqrt{2}}{2},\ \frac{-1-\sqrt{2}}{2}\right)と線分\left(\frac{1}{2}-\frac{-1-\sqrt{2}}{2},\ -\frac{1}{2}-\frac{1+\sqrt{2}}{2}\right)\)の比

\(\frac{2+\sqrt{2}}{2}:\sqrt{2}+1\)

$$\frac{2+\sqrt{2}}{2}\times\frac{2}{2+\sqrt{2}}:\sqrt{2}+1\times\frac{2}{2+\sqrt{2}}$$

\(\sqrt{2}+1\times\frac{2}{2+\sqrt{2}}\)

$$=\frac{2(\sqrt{2}+1)(2-\sqrt{2})}{4-2}=2\sqrt{2}-2+2-\sqrt{2}=\sqrt{2}$$

\(\frac{2+\sqrt{2}}{2}:\sqrt{2}+1\)

$$=1:\sqrt{2}$$

線分((1+√2)/2, (-1-√2)/2)と線分(1/2-(-1-√2)/2, -1/2-(1+√2)/2)の比

参考文献