Question

Cho \(\frac{a}{b}=\frac{b}{c}\).Chứng minh rằng:\(\frac{a}{c}=\frac{a^2+c^2}{b^2+c^2}\)

Answer 1

Có \(\frac{a}{b}=\frac{b}{c}\Leftrightarrow\frac{a}{c}=\frac{b}{d}\)

Đặt \(\frac{a}{c}=\frac{b}{d}=k\Rightarrow a=c.k;b=d.k\)

\(\Rightarrow a^2=c^2.k^2;b^2=d^2.k^2\)

Khi đó \(\frac{a^2+c^2}{b^2+d^2}=\frac{c^2.k^2+c^2}{d^2.k^2+d^2}=\frac{c^2.\left(k^2+1\right)}{d^2.\left(k^2+1\right)}=\frac{c^2}{d^2}=\frac{a^2}{b^2}\)

Answer 2

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

\(\frac{a^2}{b^2}=\frac{b^2}{c^2}=\frac{a^2+b^2}{b^2+c^2}=\frac{a}{b}\cdot\frac{b}{c}=\frac{a}{c}\)(tính chất dãy tỉ số bằng nhau)

=> đpcm 

p/s: b ghi sai đề :))