Question

Cho \(\dfrac{a+c}{b+d}=\dfrac{2a-c}{2b-d}\) chứng minh \(\dfrac{a}{b}=\dfrac{c}{d}\)

Answer 1

Giải:

Ta có:

\(\dfrac{a+c}{b+d}=\dfrac{2a-c}{2b-d}\)

\(\Leftrightarrow\dfrac{2a-c}{2b-d}=\dfrac{a+c}{b+d}\)

Áp dụng tính chất của dãy tỉ số bằng nhau, ta có:

\(\dfrac{2a-c}{2b-d}=\dfrac{a+c}{b+d}=\dfrac{2a-c-\left(a+c\right)}{2b-d-\left(b+d\right)}=\dfrac{2a-c-a-c}{2b-d-b-d}=\dfrac{a}{b}\)

\(\Leftrightarrow\dfrac{a+c}{b+d}=\dfrac{a}{b}=\dfrac{a+c-a}{b+d-b}=\dfrac{c}{d}\)

\(\Leftrightarrow\dfrac{a}{b}=\dfrac{c}{d}\) (đpcm)