The area of the triangle with vertices at $(-4, 1), (1, 2), (4, -3)$ is

The vertices of the triangle $ABC$ are $(2, 1)$,$(4, 3)$,and $(2, 5)$. If $D$,$E$,and $F$ are the mid-points of the sides,then the area of the triangle $DEF$ is:

$P(2, 1), Q(4, -1), R(3, 2)$ are the vertices of a triangle. If lines parallel to the opposite sides are drawn through $P$ and $R$ to intersect at $S$,then the area of $PQRS$ is:

$A$ triangular corner is cut from a rectangular piece of paper and the resulting pentagon has sides $5, 6, 8, 9, 12$ in some order. The ratio of the area of the pentagon to the area of the rectangle is

If $(1, 1)$,$(3, 4)$,$(5, -2)$,and $(4, -7)$ are the vertices of a quadrilateral,find its area.

The area (in square units) of the triangle formed by the points with polar coordinates $(1, 0)$,$(2, \frac{\pi}{3})$,and $(3, \frac{2\pi}{3})$ is: