Question

Cho tỉ lệ thức \(\dfrac{a}{b}=\dfrac{c}{d}\)

Chứng minh rằng: \(\left(a+2c\right)\left(b+d\right)=\left(a+c\right)\left(b+2d\right)\)

Answer 1

Giải:

Áp dụng tính chất dãy tỉ số bằng nhau ta có:
\(\dfrac{a}{b}=\dfrac{c}{d}=\dfrac{a+c}{b+d}\)

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

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

\(\Rightarrow\left(a+2c\right)\left(b+d\right)=\left(a+c\right)\left(b+2d\right)\left(đpcm\right)\)

Vậy...

Answer 2

Vì \(\dfrac{a}{b}=\dfrac{c}{d}\)

\(\Rightarrow\dfrac{a}{c}=\dfrac{b}{d}\)

Đặt \(\dfrac{a}{c}=\dfrac{b}{d}=k\)

\(\Rightarrow\left[{}\begin{matrix}a=ck\\b=dk\end{matrix}\right.\) (!)

Thay (!) vào đề bài:

VT = \(c\left(k+2\right).d\left(k+1\right)\left(1\right)\)

\(VP=c\left(k+1\right).d\left(k+2\right)\left(2\right)\)

Từ (1) và (2) \(\Rightarrow VT=VP\)

hay \(\left(a+2c\right)\left(b+d\right)=\left(a+c\right)\left(b+2d\right)\).