2021 April MIMC 10 Problems/Problem 23
On a coordinate plane, point denotes the origin which is the center of the diamond shape in the middle of the figure. Point has coordinate , and point , , and are formed through , , and rotation about the origin , respectively. Quarter circle (formed by the arc and line segments and ) has area . Furthermore, another quarter circle formed by arc and line segments , is formed through a reflection of sector across the line . The small diamond centered at is a square, and the area of the little square is . Let denote the area of the shaded region, and denote the sum of the area of the regions (formed by side , arc , and side ), (formed by side , arc , and side ) and sectors and . Find in the simplest radical form.
First of all, we know that . Since the area of the quarter circle is , we can get that Then, we can calculate the area of shaded region. It is made of two quarter circles and two right triangles. The total area would be .
The sum of the area of the regions (formed by side , arc , and side ), (formed by side , arc , and side ) and sectors and can be calculated by turning into a square, and subtract the extra areas. Since has length , we know that the height of the two right triangles are and the based are . . We want to also subtract the shaded quarter circle. The area is . The region enclosed by arc and length is the reflection of the previous area. The area . The region is also the reflection. Therefore, the total area is .
As a result, the ratio is .