Question

In the figure below, A, B, and C are the midpoints of

the sides of APQR, and M, N, and O are the midpoints
of the sides of AABC. The interiors of APAB, AAQC,
ABCR, and AMNO are shaded. What fraction of the
interior of APQR is shaded?

Answer 1

The shaded area is 3/16 of the interior of quadrilateral APQR, determined by calculating the areas of triangles formed by the midpoints of the sides.

Step-by-step explanation:

The question involves calculating the fraction of the interior of quadrilateral APQR that is shaded. Since A, B, and C are the midpoints of the sides of APQR, triangle ABC will be similar to APQR and its area will be one-fourth of the area of APQR because the sides are half as long.

If M, N, and O are the midpoints of the sides of triangle ABC, then triangle MNO will also be similar to triangle ABC and its area will be one-fourth of ABC's area or one-sixteenth of APQR's area.

The shaded area consists of the interiors of APAB, AAQC, ABCR, and AMNO. We can combine the areas of the four smaller triangles APAB, AAQC, and ABCR and find that their total area is the same as triangle ABC, which is one-fourth of APQR.

Then, we subtract the area of triangle AMNO (one-sixteenth of APQR) from this to find that the area of the shaded portion is one-fourth minus one-sixteenth of APQR. A simple calculation gives us the final fraction of the shaded area to be 3/16 of the whole area of APQR.