Question

Cho tỉ lệ thức : \(\frac{a}{b}=\frac{c}{d}\) . Chứng minh rằng : \(\frac{ac}{bd}=\frac{a^2+c^2}{b^2+d^2}\) 

Answer 1

Ta có
\(\frac{a}{b}=\frac{c}{d}\)
\(\Rightarrow\frac{a^2}{b^2}=\frac{c^2}{d^2}=\frac{a^2+c^2}{b^2+d^2}=\frac{ac}{bd}\)|
\(\Rightarrow dpcm\)

Answer 2

đặt \(\frac{a}{b}=\frac{c}{d}=k\) thì \(a=bk\text{ };\text{ }c=dk\text{ }\)

Ta có : \(\frac{ac}{bd}=\frac{bk.dk}{bd}=\frac{bd.k^2}{bd}=k^2\text{ }\left(1\right)\)

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

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

Answer 3

Ta có: \(\frac{a}{b}=\frac{c}{d}\)=> \(\left(\frac{a}{b}\right)^2=\left(\frac{c}{d}\right)^2=\frac{ac}{bd}\)=> \(\frac{a^2}{b^2}=\frac{c^2}{d^2}=\frac{ac}{bd}\)

Áp dụng tính chất dãy tỷ số bằng nhau, ta có:

\(\frac{a^2}{b^2}=\frac{c^2}{d^2}=\frac{ac}{bd}=\frac{a^2+c^2}{b^2+d^2}\)

=> \(\frac{ac}{bd}=\frac{a^2+c^2}{b^2+d^2}\) (dpcm)