Question

the coordinates of the vertices of a polygon are (0,3), (2,3), (2,0), (1, - 4), and (-2, - 1). what is the perimeter of the polygon to the nearest tenth of a unit? 15.8 units 16.9 units 17.8 units 18.6 units

Explanation:

Step1: Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)

Let the points be \(A(0,3)\), \(B(2,3)\), \(C(2,0)\), \(D(1, - 4)\), \(E(-2,-1)\)

Step2: Calculate \(AB\)

For \(A(0,3)\) and \(B(2,3)\), \(x_1 = 0,y_1=3,x_2 = 2,y_2 = 3\)
\(AB=\sqrt{(2 - 0)^2+(3 - 3)^2}=\sqrt{4+0}=2\)

Step3: Calculate \(BC\)

For \(B(2,3)\) and \(C(2,0)\), \(x_1 = 2,y_1 = 3,x_2=2,y_2 = 0\)
\(BC=\sqrt{(2 - 2)^2+(0 - 3)^2}=\sqrt{0 + 9}=3\)

Step4: Calculate \(CD\)

For \(C(2,0)\) and \(D(1,-4)\), \(x_1=2,y_1 = 0,x_2 = 1,y_2=-4\)
\(CD=\sqrt{(1 - 2)^2+(-4 - 0)^2}=\sqrt{1 + 16}=\sqrt{17}\approx4.123\)

Step5: Calculate \(DE\)

For \(D(1,-4)\) and \(E(-2,-1)\), \(x_1 = 1,y_1=-4,x_2=-2,y_2=-1\)
\(DE=\sqrt{(-2 - 1)^2+(-1+4)^2}=\sqrt{9 + 9}=\sqrt{18}\approx4.243\)

Step6: Calculate \(EA\)

For \(E(-2,-1)\) and \(A(0,3)\), \(x_1=-2,y_1=-1,x_2 = 0,y_2 = 3\)
\(EA=\sqrt{(0 + 2)^2+(3 + 1)^2}=\sqrt{4+16}=\sqrt{20}\approx4.472\)

Step7: Calculate the perimeter \(P\)

\(P=AB + BC+CD+DE+EA\)
\(P=2 + 3+4.123+4.243+4.472\)
\(P=17.838\approx17.8\)

Answer:

17.8 units