2021 April MIMC 10 Problems/Problem 24
One semicircle is constructed with diameter and let the midpoint of be . Construct a point on the side of segment (closer to segment than arc ) such that the distance from to is , and that is perpendicular to the diameter . Three more such congruent semicircles are formed through multiple rotations around the point . Name the endpoints of the diameters , , , , , in a circular direction from to . Another four congruent semicircles are constructed with diameters , and that the distance from the diameters to the point are less than the distance from the arcs to the point . Connect , , , , and . Find the ratio of the area of the pentagon to the total area of the shape formed by arcs , , , , , , , .
We can first draw the diagram with the instructions. Since and , we know that . We know that , we can calculate the area of triangle . . The final hard step for this problem is to solve for , or the diameter of the small circle. We can use coordinate geometry to finish this problem. Set as the origin point and as the positive direction. We then can get that since . We can then solve for , and it is easy as well because we can split it into two parts. . Since is perpendicular to , their slopes must be opposite reciprocal. Therefore, is to the right and down. Thus, the coordinate of would be . We can then use the distance formula on . . Then, we can draw a height from to and call that . . Thus, the area of the remaining shape of the shaded part would be . The whole shaded area would have a area of . We can connect the diameters to get a octagon. It is formed by four big triangles like and four small triangles like . Therefore, The area of the outer semicircles would form two big circles and two small circles. . Thus, the ratio would be .