Answer:
The perimeter is equal to [tex]P=(11+2\sqrt{5}+\sqrt{17})\ units[/tex] or [tex]P=19.59\ units[/tex]
Step-by-step explanation:
we have
The coordinates of the vertices are
R(-1,3), S(3,3), T(5,-1), and U(-2,-1)
plot the figure to better understand the problem
see the attached figure
we know that
The perimeter of a quadrilateral is the sum of its four length sides
so
[tex]P=RS+ST+TU+UR[/tex]
the formula to calculate the distance between two points is equal to
[tex]d=\sqrt{(y2-y1)^{2}+(x2-x1)^{2}}[/tex]
step 1
Find the distance RS
[tex]R(-1,3)\\S(3,3)[/tex]
substitute the values
[tex]d=\sqrt{(3-3)^{2}+(3+1)^{2}}[/tex]
[tex]d=\sqrt{(0)^{2}+(4)^{2}}[/tex]
[tex]RS=4\ units[/tex]
step 2
Find the distance ST
[tex]S(3,3)\\T(5,-1)[/tex]
substitute the values
[tex]d=\sqrt{(-1-3)^{2}+(5-3)^{2}}[/tex]
[tex]d=\sqrt{(-4)^{2}+(2)^{2}}[/tex]
[tex]ST=2\sqrt{5}\ units[/tex]
step 3
Find the distance TU
[tex]T(5,-1)\\U(-2,-1)[/tex]
substitute the values
[tex]d=\sqrt{(-1+1)^{2}+(-2-5)^{2}}[/tex]
[tex]d=\sqrt{(0)^{2}+(-7)^{2}}[/tex]
[tex]TU=7\ units[/tex]
step 4
Find the distance UR
[tex]U(-2,-1)\\R(-1,3)[/tex]
substitute the values
[tex]d=\sqrt{(3+1)^{2}+(-1+2)^{2}}[/tex]
[tex]d=\sqrt{(4)^{2}+(1)^{2}}[/tex]
[tex]UR=\sqrt{17}\ units[/tex]
step 5
Find the perimeter
[tex]P=RS+ST+TU+UR[/tex]
substitute the values
[tex]P=4+2\sqrt{5}+7+\sqrt{17}[/tex]
[tex]P=(11+2\sqrt{5}+\sqrt{17})\ units[/tex] -----> exact value
[tex]P=(11+4.47+4.12)=19.59\ units[/tex] -----> approximate value
