Question

Cho a = b + c và c = \(\frac{bd}{b-d}\) (b khác 0; d khác 0)

Chứng minh rằng \(\frac{a}{b}\) = \(\frac{c}{d}\)

Answer 1

Có: \(c=\frac{bd}{b-d}\)

\(\Rightarrow c\left(b-d\right)=bd\\ \Rightarrow cb-cd=bd\\ \Rightarrow bc=bd+cd\\ \Rightarrow bc=d\left(b+c\right)\)

Mà a = b + c

\(\Rightarrow bc=da\\ \Rightarrow\frac{a}{b}=\frac{c}{d}\)

Answer 2

Cách khác

Ta có: \(c=\frac{bd}{b-d}\)

\(\Rightarrow b-d=\frac{bd}{c}\left(c\ne0\right).\)

\(a=b+c\Rightarrow c=a-b\)

\(\Rightarrow c=\frac{bd}{b-d}=a-b\)

\(\Rightarrow bd=\left(a-b\right).\left(b-d\right)\)

\(\Rightarrow ab-ad-b^2+bd=bd\)

\(\Rightarrow a.\left(b-d\right)-b^2=0\)

\(\Rightarrow a.\frac{bd}{c}-b^2=0\)

\(\Rightarrow\frac{ad}{c}-b=0\)

\(\Rightarrow\frac{ad-bc}{c}=0\)

\(\Rightarrow ad-bc=0\)

\(\Rightarrow ad=bc\)

\(\Rightarrow\frac{a}{b}=\frac{c}{d}\left(đpcm\right).\)

Chúc bạn học tốt!