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Regarding the geometry of a real normed space $𝒳$, we mainly introduce a notion of approximate bisectrix-orthogonality on vectors $x,y\in 𝒳$ as follows:$${x\perp }_{W}y\text{if}\text{and}\text{only}\text{if}\sqrt{2}\frac{1-\epsilon }{1+\epsilon }\parallel x\parallel \parallel y\parallel \le \parallel \parallel y\parallel x+\parallel x\parallel y\parallel \le \sqrt{2}\frac{1+\epsilon }{1-\epsilon }\parallel x\parallel \parallel y\parallel .$$ We study the class of linear mappings preserving the approximately bisectrix-orthogonality ${\perp }_W$. In particular, we show that if $T:𝒳\to 𝒴$ is an approximate linear similarity, then $${x\perp }_{W}y⟹{Tx\perp }_{W}Ty(x,y\in 𝒳)$$ for any $\delta \in [0,1)$ and certain $\theta \ge 0$.