Question

pentagon ( great ) has vertices with coordinates,( g(0,0), r(-4,3), e(-1,6), a(4,6) ), and ( t(4,2) ).find the perimeter of pentagon ( great ).round your answer to the nearest hundredth.use the keypad to enter the answer in the box provided.the perimeter of pentagon ( great ) is ( square ) units.

Explanation:

Step1: Calculate the length of \(GR\)

Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). For \(G(0,0)\) and \(R(-4,3)\), \(d_{GR}=\sqrt{(-4 - 0)^2+(3 - 0)^2}=\sqrt{16 + 9}=\sqrt{25}=5\)

Step2: Calculate the length of \(RE\)

For \(R(-4,3)\) and \(E(-1,6)\), \(d_{RE}=\sqrt{(-1+4)^2+(6 - 3)^2}=\sqrt{9 + 9}=\sqrt{18}=3\sqrt{2}\approx4.24\)

Step3: Calculate the length of \(EA\)

For \(E(-1,6)\) and \(A(4,6)\), \(d_{EA}=\sqrt{(4 + 1)^2+(6 - 6)^2}=\sqrt{25+0}=5\)

Step4: Calculate the length of \(AT\)

For \(A(4,6)\) and \(T(4,2)\), \(d_{AT}=\sqrt{(4 - 4)^2+(2 - 6)^2}=\sqrt{0 + 16}=4\)

Step5: Calculate the length of \(TG\)

For \(T(4,2)\) and \(G(0,0)\), \(d_{TG}=\sqrt{(0 - 4)^2+(0 - 2)^2}=\sqrt{16+4}=\sqrt{20}=2\sqrt{5}\approx4.47\)

Step6: Calculate the perimeter

Perimeter \(P=d_{GR}+d_{RE}+d_{EA}+d_{AT}+d_{TG}=5 + 4.24+5+4+4.47 = 22.71\)

Answer:

\(22.71\)