What To Do If You Forgot This Formula (And Why You Shouldn’t) | Distance Between a Point and a Line

[@士-x7e]

Let Q(x1,y1), PQ=(x1-x0,y1-y0)
Normal vector of l is (a,b)
PQ・(a,b)=(x1-x0,y1-y0)・(a,b)=a(x1-x0)+b(y1-y0)=ax1-ax0+by1-by0=ax1+by1-ax0-by0
=-c-ax0-by0=-(ax0+by0+c) (∵ax1+by1+c=0)
PQ・(a,b)=±d √(a^2+b^2)
∴d=|ax0+by0+c|/√(a^2+b^2)

[@CornerstonesOfMath]

That is, I think, the simplest way to derive this formula!