Question

Chứng minh rằng : Nếu \(\dfrac{a}{b}\) = \(\dfrac{b}{c}\) thì \(\dfrac{a^2+b^2}{b^2+c^2}=\dfrac{a}{b}\) ( b,c ≠ o)

Answer 1

Ta có : \(\dfrac{a}{b}=\dfrac{b}{c}=k\rightarrow a=bk;b=ck\)

\(\dfrac{a^2+b^2}{b^2+c^2}=\dfrac{\left(bk\right)^2+b^2}{\left(ck\right)^2+c^2}=\dfrac{b^2k^2+b^2}{c^2k^2+c^2}=\dfrac{b^2\left(k^2+1\right)}{c^2\left(k^2+1\right)}=\dfrac{b^2}{c^2}\)Vì \(\dfrac{b^2}{c^2}=\dfrac{\left(ak\right)^2}{\left(bk\right)^2}=\dfrac{a^2k^2}{b^2k^2}=\dfrac{a^2}{b^2}\)

\(\Rightarrow\dfrac{a^2+b^2}{b^2+c^2}=\dfrac{a^2}{b^2}\) nếu \(\dfrac{a}{b}=\dfrac{b}{c}\)

Answer 2

Cách khác :V

Đặt: \(\dfrac{a}{b}=\dfrac{b}{c}=t\)

Nên: \(\dfrac{a^2}{b^2}=\dfrac{b^2}{c^2}=\dfrac{a^2+b^2}{b^2+c^2}=t^2\)

\(\dfrac{a}{b}.\dfrac{b}{c}=\dfrac{a}{c}=t^2\)