Question

find the perimeter of the trapezoid with these vertices (5,2),(5, - 3),(2,2),(-4, - 3). give an exact answer (not a decimal approximation). simplify your answer as much as possible.

Explanation:

Step1: Use the distance formula

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For the points \((5,2)\) and \((5,-3)\):
\(d_1=\sqrt{(5 - 5)^2+(-3 - 2)^2}=\sqrt{0+(-5)^2}=5\)
For the points \((5,-3)\) and \((-4,-3)\):
\(d_2=\sqrt{(-4 - 5)^2+(-3+3)^2}=\sqrt{(-9)^2+0}=9\)
For the points \((-4,-3)\) and \((2,2)\):
\(d_3=\sqrt{(2 + 4)^2+(2 + 3)^2}=\sqrt{6^2+5^2}=\sqrt{36 + 25}=\sqrt{61}\)
For the points \((2,2)\) and \((5,2)\):
\(d_4=\sqrt{(5 - 2)^2+(2 - 2)^2}=\sqrt{3^2+0}=3\)

Step2: Calculate the perimeter

The perimeter \(P\) of the trapezoid is \(P=d_1 + d_2+\ d_3+\ d_4\).
\(P=5 + 9+\sqrt{61}+3=17+\sqrt{61}\)

Answer:

\(17+\sqrt{61}\)