a parallelogram has the vertices (0,3), (3,0), (o,-3), and (-3,0). determine what type of parallelogram and find the perimeter and area

Looks like a square of side 3√2

so area = 18
perimeter = 12√2

To determine the type of parallelogram, perimeter, and area, we need to follow these steps:

Step 1: Plot the given vertices on a coordinate plane to visualize the parallelogram.

The given vertices are: (0,3), (3,0), (0,-3), and (-3,0).

Step 2: Determine the length of each side of the parallelogram.

By looking at the plotted vertices, we can calculate the length of each side using the distance formula:

Side AB: √((x2 - x1)^2 + (y2 - y1)^2)
= √((3 - 0)^2 + (0 - 3)^2)
= √(9 + 9)
= √18
= 3√2

Side BC: √((x2 - x1)^2 + (y2 - y1)^2)
= √((0 - 0)^2 + (-3 - 0)^2)
= √(0 + 9)
= √9
= 3

Side CD: √((x2 - x1)^2 + (y2 - y1)^2)
= √((0 - (-3))^2 + (0 - 0)^2)
= √(3^2 + 0)
= √9
= 3

Side DA: √((x2 - x1)^2 + (y2 - y1)^2)
= √((-3 - 0)^2 + (0 - 3)^2)
= √((-3)^2 + (-3)^2)
= √(9 + 9)
= √18
= 3√2

Step 3: Determine if the opposite sides of the parallelogram are equal in length.

Here, we can see that opposite sides AB and CD have the same length (3√2), while sides BC and DA have the same length (3).

Step 4: Determine if the opposite angles of the parallelogram are congruent.

Since opposite sides of the parallelogram are of equal length, we can infer that opposite angles are congruent.

Based on the above analysis, the given set of vertices forms a parallelogram. Specifically, it is a rectangle since all angles are right angles (90 degrees), and the opposite sides are congruent.

Step 5: Calculate the perimeter of the parallelogram.

The perimeter is the sum of all four sides.
Perimeter = AB + BC + CD + DA
Perimeter = 3√2 + 3 + 3 + 3√2
Perimeter = 6 + 6√2

Step 6: Calculate the area of the parallelogram.

The area of a parallelogram is determined by multiplying the base length by the corresponding height. In this case, the base length can be either AB or CD, and the corresponding height is the perpendicular distance between AB and CD.
Area = base × height

Since AB and CD are equal in length and perpendicular to each other, we can take AB as the base and calculate the height.

Height = DC = AB = 3√2

Area = base × height
Area = AB × DC
Area = 3√2 × 3√2
Area = 18

Thus, the type of parallelogram is a rectangle, the perimeter is 6 + 6√2, and the area is 18 square units.