Question

chứng minh rằng từ hệ thức a+b/a-b=c+d/c-d ta có hệ thức a/b=c/d

Answer 1

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

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

Answer 2

Ta có:

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

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

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

\(\Rightarrow\dfrac{a}{c}=\dfrac{b}{d}\)

\(\rightarrow\) đpcm

Chúc bạn học tốt!!!

Answer 3

Nếu:

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

\(\Leftrightarrow\left(a+b\right)\left(c-d\right)=\left(c+d\right)\left(a-b\right)\)

\(\Leftrightarrow a\left(c-d\right)+b\left(c-d\right)=c\left(a-b\right)+d\left(a-b\right)\)

\(\Leftrightarrow ac-ad+bc-bd=ac-bc+ad-bd\)

\(-ad+bc-bd=-bc+bc-bd\)

\(-ad=-bc\)

\(ad=bc\)

Ta có:

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

\(ad=bc\Leftrightarrow\dfrac{a+b}{a-b}=\dfrac{c+d}{c-d}\Leftrightarrow\dfrac{a}{b}=\dfrac{c}{d}\left(đpcm\right)\)

Answer 4

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