The image you've uploaded represents a geometric problem with three points arranged such that the distance between each pair of points is 1 meter. The task is to find the length of the pink path, which is represented by the distances $p$, $q$, and $r$, and connects the points in the figure.

The hint suggests dropping a perpendicular from the common point where the paths meet, which will allow us to use trigonometry to solve the problem. The angles between the paths are each 120°, which is key to calculating the distances using right triangle relationships.

Let me calculate the length of the pink path step by step.

First, consider that the three points form an equilateral triangle with each side equal to 1 meter. The angles between the pink path and the sides of the triangle are all 120°.

Using trigonometry and applying the formula for distances between points in such a configuration, we can solve for the total length of the path.

The length of the pink path, denoted as $p + q + r$, turns out to be $\frac{2}{\sqrt{3}}$ meters.

So, the correct answer is:

• $\frac{2}{\sqrt{3}}$.

Let me know if you'd like a more detailed breakdown of the solution or if you have any further questions!