In the following grid, all circles have the same center. It can be concluded that the area of the highlighted gray region is equal to

(a) two fifths of the area of the larger circle;

(b) three sevenths of the area of the larger circle;

(c) half of the area of the larger circle;

(d) four sevenths of the area of the larger circle;

(e) three fifths of the area of the larger circle.

![](https://cdn.mathpix.com/cropped/2024_05_01_666a1aec7e447b0a6cbeg-003.jpg?height=328&width=325&top_left_y=1047&top_left_x=1551)
The correct option is (c).

Observe that the figure is symmetric with respect to the line $r$ passing through the common center of the circles. For each gray region on one side of $r$, there is an equivalent white region on the other side of $r$, and vice versa. Therefore, the area of the gray region is equal to the area of the white region. Thus, the area of the gray region is equal to half the area of the larger circle.

![](https://cdn.mathpix.com/cropped/2024_05_01_666a1aec7e447b0a6cbeg-037.jpg?height=400&width=329&top_left_y=1959&top_left_x=1503)