G is the centroid of triangle ABC. The line drawn through G parallel to BC meets AB at P and meets AC at Q. What is the ratio of the area of triangle APQ to the area of triangle ABC?

Let AD be the median to BC, so G lies on it with AG : GD = 2 : 1 and therefore AG : AD = 2 : 3. Because PQ is parallel to BC, triangle APQ is similar to triangle ABC, and the similarity ratio is the ratio of the corresponding cevians, AG : AD = 2 : 3. Areas of similar figures go as the square of that ratio, so the answer is 4 : 9. The trap 2 : 3 quotes the length ratio itself as an area ratio, 1 : 9 squares the wrong ratio AG : GD read as 1 : 3, and 1 : 3 is the fraction of the triangle cut off by a median, a different figure entirely.