Question

Cho \(\dfrac{a}{b}=\dfrac{c}{d}\). Chứng minh rằng : (2 cách )

a) \(\dfrac{a+5b}{2c+5d}=\dfrac{2a-5b}{2c+5d}\)

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

Answer 1

a: Đặt a/b=c/d=k

=>a=bk; c=dk

\(\dfrac{2a+5b}{2c+5d}=\dfrac{2bk+5b}{2dk+5d}=\dfrac{b}{d}\)

\(\dfrac{2a-5b}{2c-5d}=\dfrac{2bk-5b}{2dk-5k}=\dfrac{b}{d}\)

Do đó: \(\dfrac{2a+5b}{2c+5d}=\dfrac{2a-5b}{2c-5d}\)

b: \(\dfrac{a^2-b^2}{a^2+b^2}=\dfrac{b^2k^2-b^2}{b^2k^2+b^2}=\dfrac{k^2-1}{k^2+1}\)

\(\dfrac{c^2-d^2}{c^2+d^2}=\dfrac{d^2k^2-d^2}{d^2k^2+d^2}=\dfrac{k^2-1}{k^2+1}\)

Do đó: \(\dfrac{a^2-b^2}{a^2+b^2}=\dfrac{c^2-d^2}{c^2+d^2}\)