Question

Cho Tỉ lệ thức: a/b=c/d. CMR: ac/bd=a^2+c^2/b^2+d^2

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Answer 1

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

\(\frac{ac}{bd}=\frac{a^2+c^2}{b^2+d^2}\)

Ta có : 

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

\(\Rightarrow\frac{ac}{bd}=\frac{m.b.m.d}{bd}=m.m=m^2\)

\(\Rightarrow\frac{a^2+c^2}{b^2+d^2}=\frac{\left(mb\right)^2+\left(md\right)^2}{b^2+d^2}=\frac{m^2\left(b^2+d^2\right)}{b^2+d^2}=m^2\)

Vì \(m^2=m^2\Rightarrow\frac{ac}{bd}=\frac{a^2+c^2}{b^2+d^2}\)