\(\dfrac{x}{8}=\dfrac{y}{3}=\dfrac{z}{10}\Rightarrow\dfrac{x^2}{64}=\dfrac{xy}{24}=\dfrac{yz}{30}=\dfrac{zx}{80}=\dfrac{xy+yz+zx}{24+30+80}=\dfrac{1206}{134}=9\)

\(\Rightarrow x^2=64.9=8^2.3^2\Rightarrow x=\pm24\)

With x=24 then y=24:8.3=9;z=30

With x=-24 then y=-9;z=-30

We have:

\(\dfrac{x}{8}=\dfrac{y}{3}=\dfrac{z}{10}\)

\(\Rightarrow\dfrac{x^2}{64}=\dfrac{xy}{24}=\dfrac{yz}{30}=\dfrac{xz}{80}\)

Apply the same sequence properties, we have:

\(\dfrac{x^2}{64}=\dfrac{xy}{24}=\dfrac{yz}{30}=\dfrac{xz}{80}=\dfrac{xy+yz+xz}{24+30+80}=\dfrac{1206}{134}=9\)

So \(x^2=9.64=576\)\(\Rightarrow x=24\)

\(\dfrac{xy}{24}=9\Rightarrow xy=24.9=216\Rightarrow y=9\)

\(\dfrac{yz}{30}=9\Rightarrow yz=270\Rightarrow z=30\)

So \(\left(x;y;z\right)=\left(24;9;30\right)\)