The area of parallelogram ABCD is $96{\text{cm}}^2$. The point P is chosen anywhere on the side DC. What is the area of triangle APB?

Triangle APB has AB for its base, and its apex P lies on DC, which is parallel to AB. So the height of the triangle is the distance between those parallels — exactly the height of the parallelogram. A triangle is ½ × base × height while the parallelogram is base × height, so the triangle is half of it: $48{\text{cm}}^2$, wherever P sits on DC. The trap $32{\text{cm}}^2$ assumes the three pieces DAP, APB and PBC are equal thirds, which they are not in general.