Question

Cho tỉ lệ thức \(\dfrac{a}{b}=\dfrac{c}{d}\). CMR: \(\dfrac{a^2-b^2}{ab}=\dfrac{c^2-d^2}{cd}\)

Answer 1

Từ \(\dfrac{a}{b}=\dfrac{c}{d}\)

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

=> \(\dfrac{a}{c}\).\(\dfrac{b}{d}=\dfrac{ab}{cd}=\dfrac{a}{c}\).\(\dfrac{a}{c}=\dfrac{b}{d}\).\(\dfrac{b}{d}\) \(=\dfrac{a^2}{c^2}=\dfrac{b^2}{d^2}\) (1)

Áp dụng t/c dãy tỉ số bằng nhau:

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

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

=> \(\dfrac{a^2-b^2}{ab}=\dfrac{c^2-d^2}{cd}\) (ĐPCM)