Question

[Thiên Sứ Tự Do]

Cho \(\frac{a}{b}=\frac{c}{d}\) chứng minh rằng :\(\frac{a.b}{c.d}=\frac{\left(a+b\right)^2}{\left(c+d\right)^2}\)

Answer 1

[Nguyễn Thị Vân Anh]

Đặt \(\frac{a}{b}=\frac{c}{d}=k\Rightarrow a=bk,c=dk\)

Ta có :

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

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

Từ (1) và (2) , ta có :\(\frac{a\times b}{c\times d}=\frac{\left(a+b\right)^2}{\left(c+d\right)^2}\)