Question

Cho 2(x+y)=5(y+z)=3(z+x). Chứng tỏ: \(\dfrac{x-y}{4}\)=\(\dfrac{y-z}{5}\)

Answer 1

\(2\left(x+y\right)=5\left(y+z\right)=3\left(z+x\right)\)

\(\Rightarrow\dfrac{2\left(x+y\right)}{30}=\dfrac{5\left(y+z\right)}{30}=\dfrac{3\left(z+x\right)}{30}\)

\(\Rightarrow\dfrac{x+y}{15}=\dfrac{y+z}{6}=\dfrac{z+x}{10}\)

Áp dụng tính chất dãy tỉ số bằng nhau ta có:

\(\dfrac{x+y}{15}=\dfrac{y+z}{6}=\dfrac{z+x}{10}=\dfrac{z+x-y-z}{10-6}=\dfrac{x-y}{4}\left(1\right)\)

\(\dfrac{x+y}{15}=\dfrac{y+z}{6}=\dfrac{z+x}{10}=\dfrac{x+y-z-x}{15-10}=\dfrac{y-z}{5}\left(2\right)\)

Từ (1) và (2) ta có đpcm