In the xy-plane, what is the area of a parallelogram with vertices (0,0), (2,0), (3,-2), and (5,-2)?

Answer 1

You should list your points in sequence going either clockwise or
counter-clockwise.
A quick sketch shows that you have a parallelogram with a base of 2 and
a height of 2
So the area is 2*2 or 4 units^2

Here is a nice formula for finding the area of any convex polygon:
starting with any point, list all the points in a column, repeating the point you started with.
area = (1/2)| (sum of downproducts - sum of up-products) |

0 0
2 0
5 -2
3 -2
0 0
area = (1/2)| (0 - 4 - 10 + 0 - (0 + 0 - 6 + 0)) |
= (1/2)| -14 + 6|
= (1/2)(8) = 4 as before

works every time, try it starting with another point