Question

find the perimeter of the trapezoid with these vertices. (3, 2), (-6, 2), (3, -6), (-1, -6) give an exact answer (not a decimal approximation), simplify your answer as much as possible.

Explanation:

Step1: Calculate the length of the first side

Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) for points \((3,2)\) and \((-6,2)\).
\(d_1=\sqrt{(-6 - 3)^2+(2 - 2)^2}=\sqrt{(-9)^2+0^2}=\sqrt{81}=9\)

Step2: Calculate the length of the second side

For points \((3,2)\) and \((3,-6)\).
\(d_2=\sqrt{(3 - 3)^2+(-6 - 2)^2}=\sqrt{0^2+(-8)^2}=\sqrt{64}=8\)

Step3: Calculate the length of the third side

For points \((3,-6)\) and \((-1,-6)\).
\(d_3=\sqrt{(-1 - 3)^2+(-6+6)^2}=\sqrt{(-4)^2+0^2}=\sqrt{16}=4\)

Step4: Calculate the length of the fourth side

For points \((-1,-6)\) and \((-6,2)\).
\(d_4=\sqrt{(-6 + 1)^2+(2 + 6)^2}=\sqrt{(-5)^2+8^2}=\sqrt{25 + 64}=\sqrt{89}\)

Step5: Calculate the perimeter

Perimeter \(P=d_1 + d_2 + d_3 + d_4\)
\(P=9+8+4+\sqrt{89}=21+\sqrt{89}\)

Answer:

\(21+\sqrt{89}\)