Geometry :: Example :: Coordinate Functions In The Euclidean Metric

Let \(l\) be a line in the Cartesian plane. If \(l\) is not a vertical line, then \(l\) has an equation of the form \(y=mx+b\). In that case define \(f:l \to \mathbb{R}\) by \(f(x,y) = x \sqrt{1+m^2}\). If \(l\) is a vertical line, then it has an equation of the form \(x=a\). In this case define \(f:l \to \mathbb{R}\) by \(f(a,y) = y\). Show that \(f\) is a coordinate function in the Euclidean metric.