Answer:
\(10+2\sqrt{109}\)
Step-by-step explanation:
A parallelogram is a quadrilateral with opposite sides that are both parallel and equal in length. Therefore, to find the perimeter of a parallelogram, we double the sum of the lengths of two adjacent sides.
To find the length of two adjacent sides of the parallelogram with vertices (-5, 1), (-5, -4), (5, -2), and (5, -7), calculate the distances between two pairs of points: (5, -2) and (5, -7), and (-5, 1) and (5, -2). To do this, we can use the distance formula:
\(\boxed{\begin{array}{l}\underline{\sf Distance \;Formula}\\\\d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\\\\\textsf{where:}\\ \phantom{ww}\bullet\;\;d\;\textsf{is the distance between two points.} \\\phantom{ww}\bullet\;\;\textsf{$(x_1,y_1)$ and $(x_2,y_2)$ are the two points.}\end{array}}\)
Calculate the distance between points (5, -2) and (5, -7):
\(d_1=\sqrt{(5-5)^2+(-7-(-2))^2}\\\\d_1=\sqrt{(0)^2+(-5)^2}\\\\d_1=\sqrt{0+25}\\\\d_1=\sqrt{25}\\\\d_1=5\)
Calculate the distance between points (-5, 1) and (5, -2):
\(d_2=\sqrt{(5-(-5))^2+(-2-1)^2}\\\\d_2=\sqrt{(5+5)^2+(-2-1)^2}\\\\d_2=\sqrt{(10)^2+(-3)^2}\\\\d_2=\sqrt{100+9}\\\\d_2=\sqrt{109}\)
Now, find the perimeter by doubling the sum of these two lengths:
\(\textsf{Perimeter}=2(d_1+d_2)\\\\\textsf{Perimeter}=2(5+\sqrt{109})\\\\\textsf{Perimeter}=10+2\sqrt{109}\)
So, the exact perimeter of the parallelogram with vertices (-5, 1), (-5, -4), (5, -2), and (5, -7) is:
\(\Large\boxed{\boxed{10+2\sqrt{109}}}\)
