Given: coordinates of a parallelogram. Find area.
A = bh
It's easiest to show by actually doing an example.
Example: (0, 0), (5, 3), (5, 7), (0, 4). The points (0, 0), (5,3) represent the base. The point (0, 4) = (m, n) -> " left top"
![enter image source here]()
Find the length of the base b using the distance formula:
d = sqrt((y_2-y_1)^2 + (x_2 - x_1)^2)
b = sqrt((3-0)^2 + (5-0)^2) = sqrt(9 + 25) = sqrt(34)
Find the height h using the perpendicular distance from a line to a point formula:
d =( |Am + Bn + C|)/sqrt(A^2 + B^2) where (m, n) is the left-top point we will use to drop a perpendicular to the line of the base b.
The line that represents the base needs to have the form: Ax + By + C = 0
Find the line for the base:
"slope" = m = (3-0)/(5-0) = 3/5
Line: y - y_1 = m ( x - x_1); " "y - 3 = 3/5 (x - 5)
Distribute:
y - 3= 3/5 x + 3/5* -5/1; " "y - 3 = 3/5 x -3
Base line: y = 3/5 x
Base line in Ax + By + C = 0 form:
3/5 x -y = 0; " " 5(3/5 x - y = 0) = 3x - 5y = 0
A = 3; " " B = -5; " " C = 0
h =( |3*0 + (-5)*4 + 0|)/sqrt(3^2 + (-5)^2) = |-20|/(sqrt(9 + 25)
h = 20/sqrt(34)
A = bh = sqrt(34) * 20/sqrt(34) = 20 " units"^2
