Geometry Questions: Arcs

[Gus]

I would appreciate some explanation on this type of questions. Thank you

[RonPurewal]

i assume you meant 'this particular question' rather than 'this type of question' (which is what you wrote), so i'll answer the question on that assumption.

there are only two arc-angle relationships that the gmat expects you to know:
inscribed angle (with the vertex on the circle itself): the arc that gets cut off by the angle has twice as many degrees as does the angle.
central angle (with the vertex at the center of the circle): the arc that gets cut off by the angle has the same number of degrees as does the angle.

in this case, angle QPR and angle PRO are both inscribed angles. they are also alternate interior angles (the 'Z angles' formed by parallel lines and a transversal), so both are 35 degrees. therefore, arc OP and arc QR are both 70 degrees each. since OPQR is a semicircle, it contains a total of 180 degrees, so arc PQ is 180 - 70 - 70 = 40 degrees.

40 degrees is 1/9 of a circle, so that arc is 1/9 of the total circumference of the circle, or (1/9)(18pi) = 2pi.

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