Question

Cho tỉ lệ thức \(\dfrac{a}{c}=\dfrac{c}{b}\) . Chứng minh rằng \(\dfrac{a^2+c^2}{b^2+c^2}=\dfrac{a}{b}\)

Answer 1

Đặt \(\dfrac{a}{b}=\dfrac{c}{b}=k\)

\(\Rightarrow a=c.k;c=b.k\)

Suy ra:

\(\dfrac{a^2+c^2}{b^2+c^2}=\dfrac{\left(c.k\right)^2+\left(b.k\right)^2}{b^2+\left(b.k\right)^2}=\dfrac{k^2.\left(c^2+b^2\right)}{b^2.\left(k^2+1\right)}\)

\(=\dfrac{k^2.\left[\left(b.k\right)^2+b^2\right]}{b^2.\left(k^2+1\right)}=\dfrac{k^2.\left[b^2.\left(k^2+1\right)\right]}{b^2.\left(k^2+1\right)}=k^2\) (1)

\(\dfrac{a}{b}=\dfrac{c.k}{b}=\dfrac{b.k^2}{b}=k^2\) (2)

Từ (1) và (2) \(\Rightarrow\dfrac{a^2+c^2}{b^2+c^2}=\dfrac{a}{b}\)

Chúc học tốt!!

Answer 2

đề sai òi - . -