AHSME 1974 Problem 11

If $(a, b)$ and $(c, d)$ are two points on the line whose equation is $y=mx+k$, then the distance between $(a, b)$ and $(c, d)$, in terms of $a, c,$ and $m$ is $\mathrm{(A)\ } |a-c|\sqrt{1+m^2} \mathrm{(B) \ }|a+c|\sqrt{1+m^2} \mathrm{(C) \ } \frac{|a-c|}{\sqrt{1+m^2}}$ $\mathrm{(D) \ } |a-c|(1+m^2) \mathrm{(E) \ }|a-c|\,|m|$