Short Answer 
The area of the parallelogram is calculated as 6 square units, and the rectangle, determined using the Pythagorean theorem, has an area of 34 square units. The total area combining both shapes is 40 square units.
Step 1: Calculate the Area of the Parallelogram
To find the area of the parallelogram, use the formula: area = base ‚àöo height. The base can be measured along the vertical lines, which in this case run at x=2 and x=3, giving a height of 1 unit. The base, indicated by the vertical distance from (3,3) to (3,-3), measures 6 units. Thus, the area of the parallelogram is:
- Area = 1 * 6 = 6 square units
Step 2: Calculate the Dimensions of the Rectangle
The rectangle’s vertices do not align with the x or y-axis, so we apply the Pythagorean theorem to determine its length and width. The length (L) is calculated between the points (2,1) and (-6,-1), and the width (W) between (2,1) and (3,-3). The calculations yield:
- L = sqrt((2 – -6)^2 + (1 – -1)^2) = sqrt(8^2 + 2^2) = sqrt(64 + 4) = 2*sqrt(17)
- W = sqrt((2-3)^2 + (1 – -3)^2) = sqrt((-1)^2 + 4^2) = sqrt(1 + 16) = sqrt(17)
Step 3: Calculate the Area of the Rectangle and Total Area
The area of the rectangle is determined by multiplying its length and width together. Therefore:
- Area of the rectangle = 2*sqrt(17) * sqrt(17) = 2 * 17 = 34 square units
- Finally, to get the total area, sum the area of the rectangle and the parallelogram: 34 + 6 = 40 square units.