線分((4-√2)/14, (-4+√2)/14)と線分(1/4-(4-√2)/14, -1/4-(-4+√2)/14)の比

線分\(\left(\frac{4-\sqrt{2}}{14},\ \frac{-4+\sqrt{2}}{14}\right)と線分\left(\frac{1}{4}-\frac{4-\sqrt{2}}{14},\ -\frac{1}{4}-\frac{-4+\sqrt{2}}{14}\right)\)の比

線分\(\left(\frac{4-\sqrt{2}}{14},\ \frac{-4+\sqrt{2}}{14}\right)と線分\left(\frac{1}{4}-\frac{4-\sqrt{2}}{14},\ -\frac{1}{4}-\frac{-4+\sqrt{2}}{14}\right)\)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\(\left(\frac{4-\sqrt{2}}{14},\ \frac{-4+\sqrt{2}}{14}\right)\)の距離

\(\sqrt{\left(\frac{4-\sqrt{2}}{14}\right)^2+\left(\frac{-4+\sqrt{2}}{14}\right)^2}\)

$$=\sqrt{\frac{16+2-8\sqrt{2}}{196}+\frac{16+2-8\sqrt{2}}{196}}=\sqrt{\frac{36-16\sqrt{2}}{196}}=\frac{\sqrt{36-2\sqrt{128}}}{14}$$

$$=\frac{\sqrt{32}-\sqrt{4}}{14}=\frac{4\sqrt{2}-2}{14}=\frac{2\sqrt{2}-1}{7}$$

線分\(\left(\frac{1}{4}-\frac{4-\sqrt{2}}{14},\ -\frac{1}{4}-\frac{-4+\sqrt{2}}{14}\right)\)の距離

\(\frac{1}{4}-\frac{4-\sqrt{2}}{14}\)

$$=\frac{7-8+2\sqrt{2}}{28}=\frac{-1+2\sqrt{2}}{28}$$

\(-\frac{1}{4}-\frac{-4+\sqrt{2}}{14}\)

$$=\frac{-7+8-2\sqrt{2}}{28}=\frac{1-2\sqrt{2}}{28}$$

\(\sqrt{\left(\frac{1}{4}-\frac{4-\sqrt{2}}{14}\right)^2+\left(-\frac{1}{4}-\frac{-4+\sqrt{2}}{14}\right)^2}\)

$$=\sqrt{\frac{1+8-4\sqrt{2}}{784}+\frac{1+8-4\sqrt{2}}{784}}=\sqrt{\frac{18-8\sqrt{2}}{784}}=\frac{\sqrt{18-2\sqrt{32}}}{28}=\frac{\sqrt{16}-\sqrt{2}}{28}=\frac{4-\sqrt{2}}{28}$$

線分\(\left(\frac{4-\sqrt{2}}{14},\ \frac{-4+\sqrt{2}}{14}\right)と線分\left(\frac{1}{4}-\frac{4-\sqrt{2}}{14},\ -\frac{1}{4}-\frac{-4+\sqrt{2}}{14}\right)\)の比

\(\frac{2\sqrt{2}-1}{7}:\frac{4-\sqrt{2}}{28}\)

$$\frac{2\sqrt{2}-1}{7}\times\frac{28}{4-\sqrt{2}}:\frac{4-\sqrt{2}}{28}\times\frac{28}{4-\sqrt{2}}$$

\(\frac{2\sqrt{2}-1}{7}\times\frac{28}{4-\sqrt{2}}\)

$$=\frac{8\sqrt{2}-4}{4-\sqrt{2}}=\frac{32\sqrt{2}+16-16-4\sqrt{2}}{14}=\frac{28\sqrt{2}}{14}=2\sqrt{2}$$

\(\frac{2\sqrt{2}-1}{7}:\frac{4-\sqrt{2}}{28}\)

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

線分((4-√2)/14, (-4+√2)/14)と線分(1/4-(4-√2)/14, -1/4-(-4+√2)/14)の比

参考文献