Question

Each of the three large squares shown 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?

The shaded areas in all three are equal.

Only the shaded areas of $I$ and $II$ are equal.

Only the shaded areas of $I$ and $III$ are equal.

Only the shaded areas of $II$ and $III$ are equal.

The shaded areas of $I, II,$ and $III$ are all different.

Solution:

In square $II,$ $1$ of the $4$ equal cells is shaded, so $\tfrac14$ of it is shaded.

Square $I$ breaks into $8$ equal triangles with $2$ shaded, and square $III$ breaks into $16$ equal triangles with $4$ shaded; each of these equals $\tfrac{2}{8} = \tfrac{4}{16} = \tfrac14.$

Since every figure has exactly $\tfrac14$ shaded, the shaded areas are all equal.

Thus, the correct answer is A .

1994 AMC 8 Problem 12

Solution:

Problem 12 in Other Years