Question

Cho a/c = b/d

Chứng minh ac/bd = a²+c²/b²+d²

Answer 1

Đặt \(\frac{a}{c}=\frac{b}{d}=k\)

\(\Rightarrow a=ck;b=dk\)

Khi đó : \(\frac{ac}{bd}=\frac{ckc}{dkd}=\frac{c^2}{d^2}\left(1\right)\)

\(\frac{a^2+c^2}{b^2+d^2}=\frac{\left(ck\right)^2+c^2}{\left(dk\right)^2+d^2}=\frac{c^2.k^2+c^2}{d^2.k^2+d^2}=\frac{c^2.\left(k^2+1\right)}{d^2.\left(k^2+1\right)}=\frac{c^2}{d^2}\left(2\right)\)

Từ (1) và  (2) \(\Rightarrow\frac{ac}{bd}=\frac{a^2+c^2}{b^2+d^2}\left(\text{đpcm}\right)\)

Answer 2

a/c=b/d=a^2/c^2=b^2/d^2 Theo tc dãy tỉ số = nhau,ta có: a^2/c^2=b^2/d^2=a^2+b^2/c^2+d^2 (1) a/c=b/d=a/c×a/c=b/d×a/c( nhân 2vế vs a/c) =a^2/c^2=a.b/c.d (2) Từ (1);(2)=>a.c/b.d=a^2+c^2/b^2+d^2 Mk chắc chắn dug.k cho mk