The problem is about finding the electric field produced by a dipole at a very distant point from the charges, which means that being so far the three segments $r_1, r$ and $r_2$ look parallel (and so $\theta \simeq \theta'$). I'm having hard times figuring out why the segment has length $r_2 - r_1$.
Thanks in advance.
Looking at the figure I'd read this as follows: $r_1$ is the distance between $P$ and $\oplus$. $r_2$ is the distance between $P$ and $\ominus$. The line from $\oplus$ going up and right is either a line perpendicular to the "parallel" direction of the the black lines, or a big circle of radius $r_1$. At large distance the difference between these two becomes negligible. That line meets the line to $\ominus$ and splits it into two parts: one of length $r_1$ because that's how that split point is chosen, and one of length $r_2-r_1$ because that's what remains of the total length of $r_2$.