Find the area of a parallelogram given these vertices:
P1(1,2)
P2(4,4)
P3(7,5)
P4(4,3)?

Question

Violet Aldrich · Accepted Answer

Area of parallelogram is 3

Given A (1,2), B (4,4), C(3) 7,5), D (4,3)

#AB = l = sqrt((4-1)^2 + (4-2)^2) = 3.6056#

#AD = w = sqrt((4-1)^2 + (3-2)^2) = 3.1623#

#BC = w = sqrt((7-4)^2 + (5-4)^2)) = 3.1632#

#CD = l = sqrt((7-4)^2 + (5-3)^2) = 3.6056#

Slope of #AB = m = (4-2) / (4-1) = (2/3)#

Eqn of AB
#(y-2) / (4-2) = (x-1) / (4-1)#
#3y - 6 = 2x - 2#
#2x - 3y = -4# Eqn (1)

Slope of #DE = m_1 = = -(1/m) = -(3/2)#

Eqn of DE is
#(y-3) = -(3/2)*(x-4)#
#2y - 6 = -3x + 12#
#3x + 2y = 18#. Eqn (2)

Solving equations (1) & (2) we get coordinates of point E.
Coordinates of #E(46/13, 48/13)#

Length of #DE = h = sqrt((4-(46/13)^2 + (3-(48/13)^2) 0.8321#

Area of parallelogram ABCD = #l * h = 3.6056 * 0.8321 = color (purple)3#

Caroline Aucoin · Answer

Contd.....

Prerequisites : #Area of the Delta with vertices (x_1,y_1), (x_2,y_2) and (x_3,y_3) is 1/2|D|, where, D=|(x_1,y_1,1),(x_2,y_2,1),(x_3,y_3,1)|#.

Violet Aldrich · Answer

undefined To find the area of the parallelogram, you can use the formula:

Area = |(x1y2 + x2y3 + x3y4 + x4y1) - (x2y1 + x3y2 + x4y3 + x1y4)| / 2

Substitute the coordinates of the vertices into the formula:

Area = |(1*4 + 4*5 + 7*3 + 4*2) - (4*2 + 7*4 + 4*3 + 1*5)| / 2

Calculate the values:

Area = |(4 + 20 + 21 + 8) - (8 + 28 + 12 + 5)| / 2
     = |53 - 53| / 2
     = |0| / 2
     = 0 / 2
     = 0

Therefore, the area of the parallelogram with the given vertices is 0 square units.