Slopes Only

We need to include one more variant about line intersections. It turns out that if we have a line, $L_{1}$, given by point $A$ and its slope, and a second line, $L_{2}$, going through point $B$, but perpendicular to $L_{1}$, that the intersection of the two lines can be given completely in terms of the slope. But perhaps way more important: The length of the segment from $B$ to the intersection can be written in terms of the slope only!

Distance as Slope.png — Segment $\overline{BI}$ has length as shown in the figure, written in terms of only slope, $m$. Point $A=(A_{x},A_{y})$ and point $B=(B_{x},B_{y})$. $m$ is the line slope going through point $A$.

In the figure, we write the equation of the two lines. $$L_{1}:\quad y-A_{y}=m(x-A_{x})$$ $$L_{2}:\quad y-B_{y}=\left(\frac{-1}{m}\right)(x-B_{x})$$ To get their intersection, point $I$, we just solve them simultaneously. The solution result is shown in the figure. In order to get the segment length $\overline{BI}$, we use the distance formula $$\overline{BI}=\sqrt{\left(I_{x}-B_{x}\right)^{2}+\left(I_{y}-B_{y}\right)^{2}}$$ See the figure for the result.