Question

\(Cho\dfrac{a}{3}=\dfrac{c}{4}=\dfrac{b}{5},CMR:4\left(a-b\right)\left(b-c\right)=\left(a-c\right)^2\)

Answer 1

Đặt \(\dfrac{a}{3}=\dfrac{b}{4}=\dfrac{c}{5}=k\Rightarrow a=3k;b=4k;c=5k\)

\(\Rightarrow4\left(a-b\right)\left(b-c\right)=4\left(3k-4k\right)\left(4k-5k\right)\)

\(=4.\left[\left(3-4\right).k\right].\left[\left(4-5\right).k\right]\)

\(=4.\left[-k\right].\left[-k\right]=4k^2\left(1\right)\)

\(\Rightarrow\left(a-c\right)^2=\left(3k-5k\right)^2=\left[\left(3-5\right).k\right]^2=\left[-2k\right]^2=4k^2\left(2\right)\)

Từ \(\left(1\right),\left(2\right)\Rightarrow4\left(a-b\right)\left(b-c\right)=\left(a-c\right)^2\)

Vậy \(4\left(a-b\right)\left(b-c\right)=\left(a-c\right)^2\left(dpcm\right)\)