Question

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

1. \(\dfrac{a^2+c^2}{b^2+d^2}=\dfrac{ac}{bd}\) 2. \(\dfrac{a^2+c^2}{b^2+d^2}=\dfrac{a^2-c^2}{b^2-d^2}\)

Answer 1

Đặt a/b=c/d=k

=>a=bk; c=dk

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

\(\dfrac{ac}{bd}=\dfrac{bk\cdot dk}{bd}=k^2\)

Do đó; \(\dfrac{a^2+c^2}{b^2+d^2}=\dfrac{ac}{bd}\)

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

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

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