Question

CMR: \(\dfrac{a}{b}=\dfrac{c}{d}\)thì \(\dfrac{a^2+b^2}{c^2+d^2}=\dfrac{ab}{cd}\)

Answer 1

\(\dfrac{a}{b}=\dfrac{c}{d}\\ \Rightarrow\dfrac{a}{c}=\dfrac{b}{d}\\ \Rightarrow\left(\dfrac{a}{c}\right)^2=\left(\dfrac{b}{d}\right)^2=\dfrac{ab}{cd}\\ \Rightarrow\dfrac{a^2}{c^2}=\dfrac{b^2}{d^2}=\dfrac{ab}{cd}\)

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

\(\dfrac{ab}{cd}=\dfrac{a^2}{c^2}=\dfrac{b^2}{d^2}=\dfrac{a^2+b^2}{c^2+d^2}\\ \Rightarrow\dfrac{ab}{cd}=\dfrac{a^2+b^2}{c^2+d^2}\left(đpcm\right)\)

Answer 2

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

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

\(\Rightarrow\dfrac{ab}{cd}=\dfrac{a^2}{c^2}=\dfrac{b^2}{d^2}\)

\(\Rightarrow\dfrac{a^2+b^2}{c^2+d^2}\)