Question

find the perimeter of the trapezoid with these vertices.
(5,1),(5, - 4),(1,1),( - 2, - 4)
give an exact answer (not a decimal approximation).
simplify your answer as much as possible.
scratch area (not part of answer)
perimeter:

Explanation:

Step1: Find the lengths of the sides using 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 vertical side with endpoints \((5,1)\) and \((5, - 4)\):
\(d_1=\sqrt{(5 - 5)^2+(-4 - 1)^2}=\sqrt{0+(-5)^2}=5\)
For the side with endpoints \((5,1)\) and \((1,1)\):
\(d_2=\sqrt{(1 - 5)^2+(1 - 1)^2}=\sqrt{(-4)^2+0}=4\)
For the side with endpoints \((1,1)\) and \((-2,-4)\):
\(d_3=\sqrt{(-2 - 1)^2+(-4 - 1)^2}=\sqrt{(-3)^2+(-5)^2}=\sqrt{9 + 25}=\sqrt{34}\)
For the side with endpoints \((-2,-4)\) and \((5,-4)\):
\(d_4=\sqrt{(5+2)^2+(-4 + 4)^2}=\sqrt{7^2+0}=7\)

Step2: Calculate the perimeter

The perimeter \(P\) of the trapezoid is the sum of the lengths of its sides.
\(P=d_1 + d_2+d_3+d_4=5 + 4+\sqrt{34}+7\)

Answer:

\(16+\sqrt{34}\)