Each of the three large squares shown below is the same size. Segments that intersect the sides of the squares intersect at the midpoints of the sides. How do the shaded areas of these squares compare?
Source: Julia Robinson Mathematics Festival
Slide the lower triangle up along the diagonal until it sits next to the upper triangle. Both triangles form a square the area of which equals one fourth of the area of the large square. The shaded areas in 1 have the same area as the shaded area in 2.
Divide the shaded areas in 2 and 3 into congruent triangles. The shaded area in 2 has the same area as the shaded areas in 3 because each is made up of four congruent triangles.
The shaded areas in 1, 2, and 3 have the same area.
Answer: The shaded areas in 1, 2, and 3 have the same area.