2013 HMMT November General Round Problem 2

Plot points A, B, C at coordinates (0,0), (0,1), and (1,1) in the plane, respectively. Let S denote the union of the two line segments AB and BC. Let X_1 be the area swept out when Bobby rotates S counterclockwise 45 degrees about point A. Let X_2 be the area swept out when Calvin rotates S clockwise 45 degrees about point A. Find \frac{X_1+X_2}{2}.

Answers are checked by value, so any equivalent form is accepted: 1/2, \frac{1}{2} and 0.5 all count as the same answer.

Full credit goes to HMMT for authoring these problems. This problem is from the November 2013 contest; the official solution is available on the HMMT archive. HMMT is not affiliated with or endorsing TopsOJ in any way.