It's an official problem, so it can definitely show up on the test. I don't think you need to be an engineer to solve it, but it is a good idea to draw a diagram (something I can't easily do here, but hopefully this is easy enough to follow) :
The circle has radius 1, so its circumference is 2π. If an arc has length 2π/3, it is 1/3 of the circumference, so the angle at the center of the circle will be 1/3 of 360 degrees, or 120 degrees.
So if we draw the triangle XOZ, where O is the center of the circle, we'll get an isosceles triangle, where two sides are radii of the circle and have length 1. We have a 120 degree angle at the center of the circle, so the other two equal angles must be 30 degrees.
So really the question is: if you have a 30-30-120 triangle with two sides of length 1, what is the length of the third (long) side? Since we don't know anything about 120 degree angles, we'll want to divide that angle up: if we draw a height, using the long side as our base, we'll divide the 30-30-120 triangle into two identical 30-60-90 triangles. In any 30-60-90 triangle, the sides are in a 1 to √3 to 2 ratio. Here we know the hypotenuse of each 30-60-90 is just 1, which is half of 2, so the sides of each 30-60-90 here are 1/2, √3/2 and 1 (cutting all the lengths in the 30-60-90 in half). √3/2 is the length opposite the 60 degree angle, and that's half of what we're asked to find, so the answer is √3.
You can also guess well here. If you can see you want to find the length of a proper chord of a circle of diameter 2, you can already rule out every answer that is 2 or larger, so that alone lets you guess between A and B. If you also notice the length we want belongs to a triangle with 30-30-120 angles, you can very confidently guess √3, because there's no reason a √2 would ever show up in a question with angles like this -- we see √2 when we have 45-45-90 triangles, not when we have 30 or 60 degree angles.
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