Question

[Cuc Kac]

Cho tỉ lệ thức \(\frac{x}{z}=\frac{z}{y}\). Cm \(\frac{x^2+z^2}{y^2+z^2}=\frac{x}{y}\)

Answer 1

[Thu Thao]

Đặt \(\frac{x}{z}=\frac{z}{y}=k\)

⇒ \(\left\{{}\begin{matrix}x=zk\\z=yk\end{matrix}\right.\)

Khi đó

\(\frac{x^2+z^2}{y^2+z^2}=\frac{\left(zk\right)^2+\left(yk\right)^2}{y^2+z^2}=\frac{k^2z^2+k^2y^2}{y^2+z^2}=\frac{k^2\left(z^2+y^2\right)}{y^2+z^2}=k^2\)

\(\frac{x}{y}=\frac{zk}{y}=\frac{ykk}{y}=k^2\)

Do đó \(\frac{x^2+z^2}{y^2+z^2}=\frac{x}{y}\left(=k^2\right)\)