To find the area of the parallelogram with vertices [tex]\( A(-12, 2) \)[/tex], [tex]\( B(6, 2) \)[/tex], [tex]\( C(-2, -3) \)[/tex], and [tex]\( D(-20, -3) \)[/tex], we need to follow a series of steps:
1. Determine the length of the base:
Given the vertices [tex]\( A \)[/tex] and [tex]\( B \)[/tex] lie on the same horizontal line ([tex]\(y = 2\)[/tex]), we can find the length of the base by calculating the distance between points [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
[tex]\[
\text{Base length} = |B_x - A_x| = |6 - (-12)| = |6 + 12| = 18
\][/tex]
2. Determine the height:
Since [tex]\( A \)[/tex] and [tex]\( C \)[/tex] lie on different horizontal lines ([tex]\( y = 2 \)[/tex] and [tex]\( y = -3 \)[/tex]), the height of the parallelogram is the vertical distance between these lines.
[tex]\[
\text{Height} = |A_y - C_y| = |2 - (-3)| = |2 + 3| = 5
\][/tex]
3. Calculate the area:
The formula for the area of a parallelogram is given by:
[tex]\[
\text{Area} = \text{base} \times \text{height}
\][/tex]
Substituting the values:
[tex]\[
\text{Area} = 18 \times 5 = 90 \, \text{square units}
\][/tex]
Therefore, the area of the parallelogram is
[tex]\[
\boxed{90} \, \text{units}^2
\][/tex]
