Answer:
b = 12
c = 8
d = 16
Step-by-step explanation:
The area of a parallelogram is the base times the height. If we say AD is the base, then the height is the perpendicular distance from B to AD.
First, find the equation of AD.
m = (2 − (-2)) / (4 − (-2))
m = 2/3
y − (-2) = 2/3 (x − (-2))
y + 2 = 2/3 (x + 2)
y + 2 = 2/3 x + 4/3
3y + 6 = 2x + 4
-2x + 3y + 2 = 0
The perpendicular distance between the line -2x + 3y + 2 = 0 and the point (2, b) is:
h = |-2×2 + 3b + c| / √((-2)² + 3²)
h = (-4 + 3b + c) / √13
The length of the base, AD, is:
AD = √((4 − (-2))² + (2 − (-2))²)
AD = √(36 + 16)
AD = √52
AD = 2√13
Using the area of the parallelogram:
80 = (2√13) (-4 + 3b + c) / √13
80 = 2 (-4 + 3b + c)
40 = -4 + 3b + c
44 = 3b + c
Next, use the fact that opposite sides are parallel.
ABₓ = CDₓ
2 − (-2) = c − 4
4 = c − 4
c = 8
Plug in to find b:
44 = 3b + 8
36 = 3b
b = 12
Now find d:
ABᵧ = CDᵧ
b − (-2) = d − 2
12 + 2 = d − 2
d = 16
