To find the perimeter of the parallelogram formed by the given coordinates \( A(-5,-4) \), \( B(2,-4) \), \( C(-3,2) \), and \( D(4,2) \), we first need to compute the lengths of its sides.
We can calculate the lengths of sides \( AB \) and \( BC \) (or any other two adjacent sides) using the distance formula, which is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Step 1: Calculate \( AB \)
The coordinates of points \( A \) and \( B \) are:
- \( A(-5,-4) \)
- \( B(2,-4) \)
Using the distance formula:
\[ AB = \sqrt{(2 - (-5))^2 + (-4 - (-4))^2} = \sqrt{(2 + 5)^2 + (0)^2} = \sqrt{7^2} = 7 \]
Step 2: Calculate \( BC \)
The coordinates of points \( B \) and \( C \) are:
- \( B(2,-4) \)
- \( C(-3,2) \)
Using the distance formula:
\[ BC = \sqrt{(-3 - 2)^2 + (2 - (-4))^2} = \sqrt{(-5)^2 + (6)^2} = \sqrt{25 + 36} = \sqrt{61} \]
Step 3: Using properties of parallelograms
In a parallelogram, opposite sides are equal, so:
- \( CD = AB = 7 \)
- \( AD = BC = \sqrt{61} \)
Step 4: Calculate the perimeter
The perimeter \( P \) of a parallelogram is given by:
\[ P = 2(AB + BC) \]
Substituting in the values we found:
\[ P = 2(7 + \sqrt{61}) \approx 2(7 + 7.81) \approx 2 \times 14.81 = 29.62 \text{ units} \]
Conclusion
The perimeter of the parallelogram is approximately \( 29.62 \) units. From the choices given, none of the options match exactly. There may have been a misunderstanding in the input values or scenarios.
It's important to review if the coordinates given actually form a proper parallelogram and use proper dimensional checks if required. Otherwise, the perimeter computed from all calculations provided is \( 29.62 \) units.