Question

Cho tỉ lệ thức: (a^2+b^2) / (c^2+d^2) = a.b / c.d

CMR: a / b = c / d

Answer 1

\(\frac{a^2+b^2}{c^2+d^2}=\frac{ab}{cd}\)

\(\Leftrightarrow\left(a^2+b^2\right)cd=\left(c^2+d^2\right)ab\)

\(\Leftrightarrow a^2cd-c^2ab-d^2ab+b^2cd=0\)

\(\Leftrightarrow ac\left(ad-bc\right)-bd\left(ad-bc\right)=0\)

\(\Leftrightarrow\left(ac-bd\right)\left(ad-bc\right)=0\)

\(\Leftrightarrow\begin{cases}ac=bd\\ad=bc\end{cases}\)

\(\Leftrightarrow\begin{cases}\frac{a}{b}=\frac{c}{d}\\\frac{a}{b}=\frac{d}{c}\end{cases}\)