# Mock Geometry AIME 2011 Problems/Problem 1

## Problem

Let be a unit square, and let be its image after a degree rotation about point The area of the region consisting of all points inside at least one of and can be expressed in the form where are positive integers, and shares no perfect square common factor with . Find

## Solution

Denote the intersection of and as . Also, denote a polygon enclosed in square brackets as the area as that polygon. (For example, denotes the area of polygon .)

The area of the union of the two squares is equal to . Note that , so we wish to find .

Draw in . Notice that , since , , and , so the triangles are congruent by HL. Thus, their areas are equal, and we need to find .

Note that , since that's the angle of rotation, so . Also, since , we have .

Now we have . Finally, , and our answer is , and .