Question

Cho\(\frac{a}{c}=\frac{c}{b}\) CMR: \(\frac{a^2+c^2}{b^2+c^2}=\frac{a}{b}\)

Answer 1

Đặt \(\frac{a}{c}=\frac{c}{b}=k\)

\(\Rightarrow k=\frac{a}{c}=\frac{c}{b}\Rightarrow k^2=\frac{a}{c}.\frac{c}{b}=\frac{a}{b}\left(1\right)\)

Từ \(k=\frac{a}{c}=\frac{c}{b}\Rightarrow k^2=\frac{a^2}{c^2}=\frac{c^2}{b^2}\)

Áp dụng tc dãy ti số bằng nhau ta có:

\(k^2=\frac{a^2}{c^2}=\frac{c^2}{b^2}=\frac{a^2+c^2}{c^2+b^2}\left(2\right)\)

Từ (1), (2)\(\Rightarrow\frac{a^2+c^2}{b^2+c^2}=\frac{a}{b}\left(đpcm\right)\)